I'm a retired mathematics teacher for over 35 years and have never seen or used -2^2 but have seen (-2)^2 and 3-2^2 where, in this case, the negative sign is actually a subtract sign. Its ambiguous whether that sign means a negative number or something else.
I disagree with you. if the equation says 3 - 2^2, then we will have to split the - sign from 2^2 and it will be 3 - 4, but the equation says 3 x -2^2, we don't have to split - sign from 2 because it's -2 itself raised to power 2, it will surely give us 3 x 4 which is +12, so therefore, square root of 12 x -12 is equal to 12i.
That depends. In pure math the unary and binary "-" are at the same level, so "^" is first, but many programming languages elevate unary above "^". In Octave and Matlab the unary is above the binary but still below "^" so this particular expression works like classic math. If the language's instructions aren't handy just use parens to be sure.
-2² does not mean "calculate the square of -2". It means "take the negative of 2²". We are not squaring a negative number. We are squaring a positive number and then taking the negative of the result.
@@antilogismHave you got any examples of programming languages that elevate unary - above ^? The only place I think I've come across that is Microsoft Excel, where it's a known gripe (and other similar spreadsheet applications which replicate Excel's behaviour).
@@gavindeane3670 I think I overstated it. BC is one. I'm pretty sure I stumbled on at least another. These days I mostly use octave and it's normal. I just tried awk, python3, fortran and basic and I know ada doesn't and C uses a function so it doesn't really count.
Great video I hate when you’re right that’s exactly where I made my mistake. I think it’s great how you threw in the exponent to square root equivalent which as you pointed out is the rule which allows us to go into the square root in the first place. It really being a parentheses. And thank for reminding us you can multiply the exponent in a parentheses when it’s fractional.
( (-12)^2)^(1/2) -- If we multiply the exponents we get (-12)^1 = -12. Raising a value to the power of 1/2 is not the same as finding its square root because the square root is always non-negative by definition.
When you convert equations to reverse Polish notation the order of operations becomes explicit, which is why a lot of software uses this approach. But have you ever tried to simplify such an expression?
Yes, of course! I did that in my head first , though I admit I struggled with the simple math and eventually wrote it out to check myself. 2 squared means 2 squared period! And THEN the result is a minus number, because the EXPRESSION 2 squared (2 squared) is a minus number. If it was meant to be seen as -2 (that quantity squared), it would (or should!) have been written as (-2)squared.
I just subscribed. My last algebra class was 1982 and I got it WRONG. lol. I, for some unknown reason, knew the absolute value. What I did wrong was thinking the "X" was an x whereas you meant it to be "times". That was underhanded.
-2 or any number like -a can be expressed as -1•a or in this case -1•2. You can factor out a 1 from any number. Now -2^2 can be expressed as -1•2^2. Now simply work the exponent first -1•4 and finally -4.
If you're going to explain -2² in terms of thinking of -a as -1•a, you need to explain why you're writing -2² as -1•2² not (-1•2)². In other words, you need to explain why a in this case is 2² and not 2. In general, when you take a part of a mathematical expression and replace it with something else that is equivalent, you must put the replacement inside parentheses. For example, 4/-2 is not 4/-1•2, it is 4/(-1•2). In specific cases, those parentheses would be redundant. For example 1 + -2 can be rewritten as 1 + -1•2 without parentheses. But writing it without parentheses is something that must be assessed on a case by case basis, with reference to the context in which the -2 appears. You are correct in this case. Parentheses around the -1•2 works not just be redundant, they would be incorrect. But you didn't explain why that is (which is because the - in -2² is an operator not a sign indicator, and it has lower precedence than the exponentiation).
10/12/2024 Side note : The absolute value of |-3| = 3. Watch out for this : -2² = -4 , ( - 2)² = 4 √( |-3| x -2² x (8 - 20) ) √( 3 x -2² x (-12) ) --> difference of the two terms inside the inner parentheses first √( 3 x -4 x (-12) ) --> Then the exponent √( 3 x -4 x -12 ) √(144) = ±12 --> Result or just 12 , lol
12 Edit,, There are many 'implied' parenthesis' and this problem illustrates 4 cases of parenthesis or implied The lines that bracket for 'absolute value' are a type of parenthesis ,, the -2 squared is another the implication is that the number 2 squared as a total is squared,, there is a true parenthesis and then the square root symbol that encompasses the entire problem is another implied parenthesis. And just as you showed,,, with TH-cam shoving 6 commercials down my throat every 1.75 minutes Resolve the the three separate items under the square root symbol and then you can tackle that all encompassing square root, One set of parenthesis and three implied parenthesis.
@@Sailor376also Your analysis is spot on. Just to note that the correct term you're looking for is "grouping symbol". "Parentheses" specifically means these (). The only place actual parentheses are implied is around the 2² in -2². The absolute value symbol and the horizontal line in the square roots symbol don't imply parentheses or anything. They are themselves, explicitly, grouping symbols.
@@gavindeane3670 Thank you !!!! As you may have guessed, I am not trained in mathematics. Most of what I do is quite intuitive, but I do not have the words always. So a boost from you,, much appreciated.
@@Sailor376alsoNo worries. Another thing to note is about the acronym PEMDAS that he talks about in the video. Often the acronym is modified to GEMDAS, with G for Groupings, to acknowledge the fact that parentheses are not the only grouping symbol in mathematics.
@@gavindeane3670 I'll watch for that ! When I do bump into it, it will be no surprise. Thanks. You'd laugh at me.. an idiot savant possibly,, heavy emphasis on the idiot part. I have a building going up soon (contractor) 21'X30',, the corner to corner will be for squaring the concrete forms 36' 7&3/8ths" ?? I can't trust that, I'll have to check on a calculator, but it can save a ton of time walking around in the mud and rain of a jobsite to be able to double check my figures or the architect's. But,,, I let the bank balance my checking account,, laughing,, the idiot part.
This is a great example of where the teaching of mathematics goes off the rails. Mathematics is a 'language". It is a common language spoken and written the world over. The trick is to express in mathematical terms what is is you say in your own verbal communication. For example, do you mean to ask what is the result of taking the negative of 2 squared, or taking the square of negative 2. Two different meaning completely. To reduce confusion, place the negative 2 in brackets if you mean the second. Language is always more useful if individual interpretation is not a factor. The key element here is what is written in English, below the mathematical expression. Does the math match the English? Yes or No. Answer that first. Poor English. Is the question (negative 2) squared, or negative (two squared)?
It's the difference between an academic approach and a real world approach. In the real world the negative in the middle term wouldn't be left there, it would be moved to the beginning of section under the radical. If the negative before the two comes from substituting values into a general equation then the negative is part of the value and should be squared. If the negative is part of the general equation then it should be moved to the front of the equation under the radical. The equation as written only exists to deliberately obfuscate the problem.
If the negative comes from substituting values into a general equation and so we're squaring a negative value, then it must be written (-2)². It's not written like that so we are not squaring a negative value.
@@gavindeane3670 I realize how it works and I'm just saying I'm the real world you will never come across an equation that contains a negative in that location. If it's properly simplified the negative will be moved in front of the absolute value of the first term.
@@gavindeane3670 Yes! the -2 squared is clearly a SUBTRACTION of an operational resultant quantity (2 squared), not a request to square a minus NUMBER (value)! (I can't type the super-text 2 for squared, so I have to type squared.)
In the (badly written) title, does the phrase "... negative 2 squared ..." mean - (2^2) or (-2)^2 ? It is completely ambiguous, and I maintain it means the latter [ (-2)^2 ] , since English does not have PEMDAS rules. Furthermore, the use of the word *negative* means it *has* to refer to the 2; it is not the same as the word *minus* here. Very badly (actually wrongly) worded - in fact it does not match the mathematical expression.
Yes! I hadn't even noticed the TITLE, just looked at the math symbols and interpreted THEM. It IS clearly NOT meant to mean 'square minus 2', but rather square 2 and then subtract it from the rest of the expression under the square-root sign. So the title is wrong, or as you say, 'badly written'.
This was a very poorly worded problem. Using the commutative property, you can rewrite what’s under the radical to: |-3| x (8-20) x -2^2, which might reduce to either 3 x (-12) x 4, or 3 x (-12) x (-4). Now, the problem was worded as “times -2 squared” (which reduces to -144), rather than “times the negative of 2 squared” (which reduces to) 144. The answer should be 12i.
I was puzzled about what you meant at first, then I realised you were talking about the textual title/description. I agree. Given the intended meaning, as evidenced by the video contents, "the negative of 2 squared" would have been better wording.
I got yet another answer because I read "x" as a variable, not an operation. PEMDAS is not for establishing superiority, it's a convention. Equations need to be clear. Math should be the puzzle, not order of operations. These days, every time I hear someone say PEMDAS is evaluated within groups left to right I shake my head. That's not the rule. Go on, PEMDAS prosecutors, tell me PEMDAS is evaluated left to right within categories. I'm ready for you! ;-)
The wording in the video description is muddled, but the expression in the video is clear. We are multiplying the factors 3, -4, and -12, and then taking the principle square route of the result.
@@gavindeane3670 Would your statement be true if you renamed the variable x in the thumbnail to y? Put another way, why did you assume x was an operator, not a variable?
That's a good question. And part of the point of this video is to teach that -2² means -(2²) not (-2)². So the answer to your question: how are you supposed to know? Mathematics teachers need to tell you - that's how you discover what -2² means.
sqrt (3 x (negative 2) sq x (8-20)) = sqrt(3 x 4 x -12) = sqrt(12 x -12) = sqrt (-144) = sqrt 12i. The "x' in front of the "-2 sq" is the arithmetic function which makes the "-" a state sign of the "2" and negative 2 squared is 4, not -4. You should not have TWO CONSECUTIVE ARITHMETIC FUNCTIONS. You are squaring a negative 2, not subtracting 2 squared. UNLESS the 2 squared is in parentheses, then the 4 would be negative. But it is not in this problem.
There's no problem with the - following the ×. We are clearly and unequivocally told that the first operand for that multiplication is the value of the expression |-3|, which is 3, and the second operand for that multiplication is the value of the expression -2², which is -4. We are certainly not subtracting. That - symbol is obviously not a subtraction operator. We are not squaring a negative number either. That - symbol is not a sign indicator in a negative number. That - symbol is, as it always is in the expression -x², a negation operator. The fact that it happens to be following a multiplication operator in this instance does not change what -x² means. If the use of two operators in succession like that grates with you a bit (and I understand why it might) then you can just put the -2² in parentheses. That's fine, but from a syntax point of view it isn't necessary.
@@gavindeane3670 Yes, "you can just put the -2² in parentheses", so it would read -(2 squared), though not (-2²), because that would still be ambiguous, suggesting square a minus 2?
@@dannichols6261 The person I was replying to didn't seem to like the - immediately following the × in |-3| × -2² That syntax is fine, but to avoid - following × you can write |-3| × (-2²) That solves the problem of - immediately following ×, if someone considers it to be a problem in the first place. Understanding that -2² means "take the negative of 2²" not "calculate the square of -2" is a completely unrelated issue.
It's just +12. The √ symbol tells you to calculate only the principal square root (that's the positive square root for positive real numbers like 144).
Ooohhhh... sorry - I really don't agree here. Negation is not subtraction. Yes, you can say -2 = (0-2) but that's not the convention. The convention is negation isn't an operator, it's representation of the number itself otherwise the naturals would only go from 0 up and we'd say there are no negative numbers - just (0-n). When a number has a 'floating' negative, it's negation, not subtraction - which is different from when it's used with a variable. It's why the correct way to say "-2" is "negative two" not "minus two".
The correct way to say the name of negative numbers is just a convention of language. I think it might be regional. Watching videos online presented by Americans, I get the impression that the usual convention in that country is to call the negative number -2 "negative two". Here in the UK that number is routinely called "minus two". As for the notation, the convention with the notation -2² absolutely IS that the - symbol is a negation operator. It is not treated as a character being used to write the negative number "-2". It is an operator being applied to the expression 2².
Checking several Google entries for minus two squared I found both positive and negative results. And then I found a definitive that it's ambiguous as written and REQUIRES parentheses to resolve the ambiguity. Hence I would say your problem has no solution as it is improperly written. Please cite a valid authority of why your way of writing and interpreting it is correct.
Can you give a reference to any authority that says it is ambiguous? Or that it definitely means (-2)²? The easy way to understand why we treat -2² as -(2²) rather than (-2)² is to think what would happen if we did it the other way around. The convention we use is to treat the - as a negation operator rather than a sign indicator in a negative number. This keeps the notation consistent with other examples like -x² and -(1+1)² where the - symbol cannot be anything other than a negation operator. If it worked the other way round then the answers to the questions "What is -2²?" and "What is -x² for x=2?" would not be the same. Which would be a bit weird. There's also commutativity, which tells us that a - b is the same as -b + a. But if -2² meant "calculate the square of -2" then 1 - 2² and -2² + 1 would not be the same.
No. The √ symbol is more precisely defined than that. It refers specifically to the principal square root of a number (that is, the positive square root when we're dealing with positive real numbers like this). √(144) equals 12 and only 12. If you want to refer to both square roots you write ±√(144).
@@gavindeane3670 - Thanks . I was never aware until I just googled it. "It's called the principal square root function, which cannot give a negative result."
"Negative" isn't an operator. It's a sign. "Minus" is an operator. They share a symbol, with context providing the meaning. "Thee times the negative square of two" is a perfectly valid concept.
Just for fun, I asked Microsoft's Copilot, and it got really confused: Me, using the exact wording in the title of the video: what is "square root of (|- 3| times negative 2 squared times (8 minus 20))" Copilot: Alright, let's break it down step by step: Absolute value of -3 is |-3|, which is 3. Negative 2 squared is (-2)^2, which is 4. 8 minus 20 is (8 - 20), which is -12. Now, multiply these results together: 3 × 4 × (-12) = 3 × 4 × (-12) = 144. Finally, we find the square root of 144, which is 12. So, the answer is 12. Pretty cool, right? Me: So, are you saying that 3 x 4 x -12 is 144? Copilot: Eek, good catch. No, my math was off. Let's fix it: Absolute value of -3 is |-3|, which is 3. Negative 2 squared is (-2)^2, which is 4. 8 minus 20 is (8 - 20), which is -12. Now, multiplying these: 3 × 4 × (-12) = 3 × 4 × (-12) = -144. Finally, we find the square root of -144. But here's the snag: you can't take the square root of a negative number in the realm of real numbers. In the complex world, the answer would involve imaginary numbers. So, thanks to you, math geekage!
Oh, dear, another rocky landing for PEMDAS. Absolute value isn't in PEMDAS, but we assume it is like a parenthetical multiplication by one or minus one as appropriate. I read this as square root of |-3|x - 2^2x(8-20), or 3x - 32x - 80x, or square root of 109x. If you leave ambiguity in your expressions, no matter how effectively you can prosecute your case in PEMDAS court, you are not communicating effectively.
In the order of operations (aka "pemdas", "bodmas" or whatever) the absolute value symbols are treated as grouping symbols. So yes, absolute value definitively is "in pemdas." And the × is a multiplication operator, not a letter "x" being used as a variable. For comparison: ×x. Note that the centre of the former is at the same height as the other operators (×÷+−) while the letter sits lower.
The absolute value symbol is an example of why PEMDAS is extended to GEMDAS with G for Groupings. Indeed, the long vinculum on the √ symbol is another example of that.
@@dazartingstall6680 How nice. When I was in school I had undiagnosed visual impairment. That fine a visual distinction would have been easily lost to me. In other posts, when I tried to point out clarity is not a sin by using spaces to illustrate ambiguity, I was chastised for assuming spacing meant anything. PEMDAS by any name is necessary. All I ask is that math be communicated clearly. In another video in this channel, there was an expression something like 5/6x. PEMDAS says that's (5/6)x. To my eye, those parentheses don't add clutter, they derail misreading that as 5/(6x) - which it could have been if 6x had been under a horizontal line instead of a diagonal. Parentheses can be overused. PEMDAS can be overused, too.
If the absolute number is just the positive of that number, why (ex. absolute number of -3 = +3 ) why do we even have a term for it. Why isn't it just written as a positive number in the first place. eliminating a need for the absolute symbol all together. I know I'm showing my ignorance with this question, but ignorance is curable, it's stupidity thar isn't.
We don't only do mathematics with numbers. Often we're working with generalised equations and formulae using algebra. And in that case, it can be useful to have a way to say "the magnitude of this variable" as distinct from its actual value. A simple example of how this can be used is √(x²) = |x|
Thank you. I learnt two things, that the [absolute number] = value of distance from 0 and the difference between -2 squared = -4 and (-2) squared = 4.
I'm a retired mathematics teacher for over 35 years and have never seen or used -2^2 but have seen (-2)^2 and 3-2^2 where, in this case, the negative sign is actually a subtract sign. Its ambiguous whether that sign means a negative number or something else.
For clarification purposes, there should be parentheses enclosing the entire-2^2 expression. The result would still be -4.
I disagree with you. if the equation says 3 - 2^2, then we will have to split the - sign from 2^2 and it will be 3 - 4, but the equation says 3 x -2^2, we don't have to split - sign from 2 because it's -2 itself raised to power 2, it will surely give us 3 x 4 which is +12, so therefore, square root of 12 x -12 is equal to 12i.
That depends. In pure math the unary and binary "-" are at the same level, so "^" is first, but many programming languages elevate unary above "^". In Octave and Matlab the unary is above the binary but still below "^" so this particular expression works like classic math. If the language's instructions aren't handy just use parens to be sure.
-2² does not mean "calculate the square of -2". It means "take the negative of 2²".
We are not squaring a negative number. We are squaring a positive number and then taking the negative of the result.
@@antilogismHave you got any examples of programming languages that elevate unary - above ^?
The only place I think I've come across that is Microsoft Excel, where it's a known gripe (and other similar spreadsheet applications which replicate Excel's behaviour).
@@gavindeane3670 I think I overstated it. BC is one. I'm pretty sure I stumbled on at least another. These days I mostly use octave and it's normal. I just tried awk, python3, fortran and basic and I know ada doesn't and C uses a function so it doesn't really count.
My eyes failed me. I did not notice that that whole problem was a square root.I didn't notice the square root big box around it.
Great video I hate when you’re right that’s exactly where I made my mistake. I think it’s great how you threw in the exponent to square root equivalent which as you pointed out is the rule which allows us to go into the square root in the first place. It really being a parentheses. And thank for reminding us you can multiply the exponent in a parentheses when it’s fractional.
( (-12)^2)^(1/2) -- If we multiply the exponents we get (-12)^1 = -12. Raising a value to the power of 1/2 is not the same as finding its square root because the square root is always non-negative by definition.
When you convert equations to reverse Polish notation the order of operations becomes explicit, which is why a lot of software uses this approach. But have you ever tried to simplify such an expression?
√(|−3| × −2² × (8 − 20))
√(3 × −4 × −12)
√144
12
Yes, of course! I did that in my head first , though I admit I struggled with the simple math and eventually wrote it out to check myself. 2 squared means 2 squared period! And THEN the result is a minus number, because the EXPRESSION 2 squared (2 squared) is a minus number. If it was meant to be seen as -2 (that quantity squared), it would (or should!) have been written as (-2)squared.
I just subscribed. My last algebra class was 1982 and I got it WRONG. lol. I, for some unknown reason, knew the absolute value. What I did wrong was thinking the "X" was an x whereas you meant it to be "times". That was underhanded.
-2 or any number like -a can be expressed as -1•a or in this case -1•2. You can factor out a 1 from any number. Now -2^2 can be expressed as -1•2^2. Now simply work the exponent first
-1•4 and finally -4.
If you're going to explain -2² in terms of thinking of -a as -1•a, you need to explain why you're writing -2² as -1•2² not (-1•2)². In other words, you need to explain why a in this case is 2² and not 2.
In general, when you take a part of a mathematical expression and replace it with something else that is equivalent, you must put the replacement inside parentheses. For example, 4/-2 is not 4/-1•2, it is 4/(-1•2).
In specific cases, those parentheses would be redundant. For example 1 + -2 can be rewritten as 1 + -1•2 without parentheses. But writing it without parentheses is something that must be assessed on a case by case basis, with reference to the context in which the -2 appears.
You are correct in this case. Parentheses around the -1•2 works not just be redundant, they would be incorrect. But you didn't explain why that is (which is because the - in -2² is an operator not a sign indicator, and it has lower precedence than the exponentiation).
10/12/2024
Side note : The absolute value of |-3| = 3. Watch out for this : -2² = -4 , ( - 2)² = 4
√( |-3| x -2² x (8 - 20) )
√( 3 x -2² x (-12) ) --> difference of the two terms inside the inner parentheses first
√( 3 x -4 x (-12) ) --> Then the exponent
√( 3 x -4 x -12 )
√(144) = ±12 --> Result or just 12 , lol
12 Edit,, There are many 'implied' parenthesis' and this problem illustrates 4 cases of parenthesis or implied The lines that bracket for 'absolute value' are a type of parenthesis ,, the -2 squared is another the implication is that the number 2 squared as a total is squared,, there is a true parenthesis and then the square root symbol that encompasses the entire problem is another implied parenthesis. And just as you showed,,, with TH-cam shoving 6 commercials down my throat every 1.75 minutes Resolve the the three separate items under the square root symbol and then you can tackle that all encompassing square root,
One set of parenthesis and three implied parenthesis.
@@Sailor376also Your analysis is spot on. Just to note that the correct term you're looking for is "grouping symbol". "Parentheses" specifically means these (). The only place actual parentheses are implied is around the 2² in -2². The absolute value symbol and the horizontal line in the square roots symbol don't imply parentheses or anything. They are themselves, explicitly, grouping symbols.
@@gavindeane3670 Thank you !!!! As you may have guessed, I am not trained in mathematics. Most of what I do is quite intuitive, but I do not have the words always. So a boost from you,, much appreciated.
@@Sailor376alsoNo worries. Another thing to note is about the acronym PEMDAS that he talks about in the video. Often the acronym is modified to GEMDAS, with G for Groupings, to acknowledge the fact that parentheses are not the only grouping symbol in mathematics.
@@gavindeane3670 I'll watch for that ! When I do bump into it, it will be no surprise. Thanks. You'd laugh at me.. an idiot savant possibly,, heavy emphasis on the idiot part. I have a building going up soon (contractor) 21'X30',, the corner to corner will be for squaring the concrete forms 36' 7&3/8ths" ?? I can't trust that, I'll have to check on a calculator, but it can save a ton of time walking around in the mud and rain of a jobsite to be able to double check my figures or the architect's. But,,, I let the bank balance my checking account,, laughing,, the idiot part.
the only tricky part is the term -2^2. Can't believe it take 17 minutes to explain. but appreciate his attempts to simplify.
How long do you need to solve that simple problem?
The only equation you have to remember is length TIMES girth over yaw divided by mass equals TMI.
√ l-3l x -2^2 x (8-20)
Applying PEMDAS,
√l-3l x -2^2 x -12
√l-3l x -4 x-12
√3 x -4 x -12
√ 144
12
You made it clear Chris! I had 144. Following pemdas. I just forgot to keep the square root symbol. Thanks for taking the time to put the problem down
@@veraturner
Thank you for your kind words.
We are not perfect and we make mistakes in everything, now and then.
But, each mistake makes us humble! 😊
This is a great example of where the teaching of mathematics goes off the rails. Mathematics is a 'language". It is a common language spoken and written the world over. The trick is to express in mathematical terms what is is you say in your own verbal communication.
For example, do you mean to ask what is the result of taking the negative of 2 squared, or taking the square of negative 2. Two different meaning completely. To reduce confusion, place the negative 2 in brackets if you mean the second. Language is always more useful if individual interpretation is not a factor.
The key element here is what is written in English, below the mathematical expression. Does the math match the English? Yes or No. Answer that first. Poor English. Is the question (negative 2) squared, or negative (two squared)?
Answer is 12 if we apply PEDMAS rule
got 12 sr 3 x -4 x (-12) so sr -12 x -12 so sr 144 = 12 thanks for the fun
Got 12 after looking at the TH-cam thumbnail for 30 seconds. It's not really that complicated
It's the difference between an academic approach and a real world approach. In the real world the negative in the middle term wouldn't be left there, it would be moved to the beginning of section under the radical.
If the negative before the two comes from substituting values into a general equation then the negative is part of the value and should be squared. If the negative is part of the general equation then it should be moved to the front of the equation under the radical.
The equation as written only exists to deliberately obfuscate the problem.
If the negative comes from substituting values into a general equation and so we're squaring a negative value, then it must be written (-2)². It's not written like that so we are not squaring a negative value.
@@gavindeane3670 I realize how it works and I'm just saying I'm the real world you will never come across an equation that contains a negative in that location. If it's properly simplified the negative will be moved in front of the absolute value of the first term.
@@gavindeane3670 Yes! the -2 squared is clearly a SUBTRACTION of an operational resultant quantity (2 squared), not a request to square a minus NUMBER (value)! (I can't type the super-text 2 for squared, so I have to type squared.)
Thank you
sqrt 144 = 12
12...😊
V(3 x -4 x -12) = V(-12 x -12) = V144 = 12 I just popped a braincell to solve this...
In the (badly written) title, does the phrase "... negative 2 squared ..." mean - (2^2) or (-2)^2 ? It is completely ambiguous, and I maintain it means the latter [ (-2)^2 ] , since English does not have PEMDAS rules. Furthermore, the use of the word *negative* means it *has* to refer to the 2; it is not the same as the word *minus* here. Very badly (actually wrongly) worded - in fact it does not match the mathematical expression.
Yes! I hadn't even noticed the TITLE, just looked at the math symbols and interpreted THEM. It IS clearly NOT meant to mean 'square minus 2', but rather square 2 and then subtract it from the rest of the expression under the square-root sign. So the title is wrong, or as you say, 'badly written'.
mentor me to start my own channel like this.
12
What real life situation does this equation solve? PEMDAQ.
This was a very poorly worded problem. Using the commutative property, you can rewrite what’s under the radical to: |-3| x (8-20) x -2^2, which might reduce to either 3 x (-12) x 4, or 3 x (-12) x (-4). Now, the problem was worded as “times -2 squared” (which reduces to -144), rather than “times the negative of 2 squared” (which reduces to) 144. The answer should be 12i.
That’s the same answer I got. He needs to be more clear next time.
I was puzzled about what you meant at first, then I realised you were talking about the textual title/description. I agree. Given the intended meaning, as evidenced by the video contents, "the negative of 2 squared" would have been better wording.
I got yet another answer because I read "x" as a variable, not an operation.
PEMDAS is not for establishing superiority, it's a convention. Equations need to be clear. Math should be the puzzle, not order of operations.
These days, every time I hear someone say PEMDAS is evaluated within groups left to right I shake my head. That's not the rule. Go on, PEMDAS prosecutors, tell me PEMDAS is evaluated left to right within categories. I'm ready for you! ;-)
The wording in the video description is muddled, but the expression in the video is clear. We are multiplying the factors 3, -4, and -12, and then taking the principle square route of the result.
@@gavindeane3670 Would your statement be true if you renamed the variable x in the thumbnail to y? Put another way, why did you assume x was an operator, not a variable?
Ok so without parenthesis how am i supposed to know if it's -(2^2) or (-2)^2?
That's a good question. And part of the point of this video is to teach that -2² means -(2²) not (-2)².
So the answer to your question: how are you supposed to know? Mathematics teachers need to tell you - that's how you discover what -2² means.
sqrt (3 x (negative 2) sq x (8-20)) = sqrt(3 x 4 x -12) = sqrt(12 x -12) = sqrt (-144) = sqrt 12i. The "x' in front of the "-2 sq" is the arithmetic function which makes the "-" a state sign of the "2" and negative 2 squared is 4, not -4. You should not have TWO CONSECUTIVE ARITHMETIC FUNCTIONS. You are squaring a negative 2, not subtracting 2 squared. UNLESS the 2 squared is in parentheses, then the 4 would be negative. But it is not in this problem.
There's no problem with the - following the ×. We are clearly and unequivocally told that the first operand for that multiplication is the value of the expression |-3|, which is 3, and the second operand for that multiplication is the value of the expression -2², which is -4.
We are certainly not subtracting. That - symbol is obviously not a subtraction operator. We are not squaring a negative number either. That - symbol is not a sign indicator in a negative number.
That - symbol is, as it always is in the expression -x², a negation operator. The fact that it happens to be following a multiplication operator in this instance does not change what -x² means.
If the use of two operators in succession like that grates with you a bit (and I understand why it might) then you can just put the -2² in parentheses. That's fine, but from a syntax point of view it isn't necessary.
@@gavindeane3670 Yes, "you can just put the -2² in parentheses", so it would read
-(2 squared), though not (-2²), because that would still be ambiguous, suggesting square a minus 2?
@@dannichols6261 The person I was replying to didn't seem to like the - immediately following the × in
|-3| × -2²
That syntax is fine, but to avoid - following × you can write
|-3| × (-2²)
That solves the problem of - immediately following ×, if someone considers it to be a problem in the first place.
Understanding that -2² means "take the negative of 2²" not "calculate the square of -2" is a completely unrelated issue.
The self portraits of Mr. TH-cam Math Man scare me more than the problem solving.
As in the video, I am assuming -2sq2 is -4. The expression then works out to sqrt(-12x-12).
The answer should be -12.
Or +/- 12
-12 × -12 is 144
√(144) is 12
The answer is not -12 and it's not ±12. The √ symbol is more precisely defined than that.
12 without a calculator
I find the teaching style includes so many chatty asides and repetitions that I lose focus on the main thrust of the lesson.
+12,-12
It's just +12. The √ symbol tells you to calculate only the principal square root (that's the positive square root for positive real numbers like 144).
Ooohhhh... sorry - I really don't agree here. Negation is not subtraction. Yes, you can say -2 = (0-2) but that's not the convention. The convention is negation isn't an operator, it's representation of the number itself otherwise the naturals would only go from 0 up and we'd say there are no negative numbers - just (0-n). When a number has a 'floating' negative, it's negation, not subtraction - which is different from when it's used with a variable. It's why the correct way to say "-2" is "negative two" not "minus two".
The correct way to say the name of negative numbers is just a convention of language. I think it might be regional. Watching videos online presented by Americans, I get the impression that the usual convention in that country is to call the negative number -2 "negative two". Here in the UK that number is routinely called "minus two".
As for the notation, the convention with the notation -2² absolutely IS that the - symbol is a negation operator. It is not treated as a character being used to write the negative number "-2". It is an operator being applied to the expression 2².
Checking several Google entries for minus two squared I found both positive and negative results. And then I found a definitive that it's ambiguous as written and REQUIRES parentheses to resolve the ambiguity. Hence I would say your problem has no solution as it is improperly written. Please cite a valid authority of why your way of writing and interpreting it is correct.
Can you give a reference to any authority that says it is ambiguous? Or that it definitely means (-2)²?
The easy way to understand why we treat -2² as -(2²) rather than (-2)² is to think what would happen if we did it the other way around.
The convention we use is to treat the - as a negation operator rather than a sign indicator in a negative number. This keeps the notation consistent with other examples like -x² and -(1+1)² where the - symbol cannot be anything other than a negation operator.
If it worked the other way round then the answers to the questions
"What is -2²?"
and
"What is -x² for x=2?"
would not be the same. Which would be a bit weird.
There's also commutativity, which tells us that a - b is the same as -b + a. But if -2² meant "calculate the square of -2" then 1 - 2² and -2² + 1 would not be the same.
144
Could the answer be +12 OR -12………?
No. The √ symbol is more precisely defined than that. It refers specifically to the principal square root of a number (that is, the positive square root when we're dealing with positive real numbers like this).
√(144) equals 12 and only 12.
If you want to refer to both square roots you write ±√(144).
Strictly , the answer should be +/- 12
No it shouldn't. The √ symbol is more precisely defined than that.
√144 is 12
±√144 is ±12
@@gavindeane3670 - Thanks . I was never aware until I just googled it. "It's called the principal square root function, which cannot give a negative result."
Wow
12 in 20 seconds in my head because I just woke up. Sorry about that.
Ahhhhh, the good old "times negative" operator.
It's fine. It's two separate operators.
"Negative" isn't an operator. It's a sign. "Minus" is an operator. They share a symbol, with context providing the meaning. "Thee times the negative square of two" is a perfectly valid concept.
Just for fun, I asked Microsoft's Copilot, and it got really confused:
Me, using the exact wording in the title of the video:
what is "square root of (|- 3| times negative 2 squared times (8 minus 20))"
Copilot:
Alright, let's break it down step by step:
Absolute value of -3 is |-3|, which is 3.
Negative 2 squared is (-2)^2, which is 4.
8 minus 20 is (8 - 20), which is -12.
Now, multiply these results together: 3 × 4 × (-12) = 3 × 4 × (-12) = 144.
Finally, we find the square root of 144, which is 12. So, the answer is 12. Pretty cool, right?
Me:
So, are you saying that 3 x 4 x -12 is 144?
Copilot:
Eek, good catch. No, my math was off. Let's fix it:
Absolute value of -3 is |-3|, which is 3.
Negative 2 squared is (-2)^2, which is 4.
8 minus 20 is (8 - 20), which is -12.
Now, multiplying these: 3 × 4 × (-12) = 3 × 4 × (-12) = -144.
Finally, we find the square root of -144. But here's the snag: you can't take the square root of a negative number in the realm of real numbers. In the complex world, the answer would involve imaginary numbers. So, thanks to you, math geekage!
PEMDAS Please Excuse My Dumb A.. Student!
Oh, dear, another rocky landing for PEMDAS. Absolute value isn't in PEMDAS, but we assume it is like a parenthetical multiplication by one or minus one as appropriate.
I read this as square root of |-3|x - 2^2x(8-20), or 3x - 32x - 80x, or square root of 109x.
If you leave ambiguity in your expressions, no matter how effectively you can prosecute your case in PEMDAS court, you are not communicating effectively.
In the order of operations (aka "pemdas", "bodmas" or whatever) the absolute value symbols are treated as grouping symbols. So yes, absolute value definitively is "in pemdas." And the × is a multiplication operator, not a letter "x" being used as a variable.
For comparison: ×x. Note that the centre of the former is at the same height as the other operators (×÷+−) while the letter sits lower.
The absolute value symbol is an example of why PEMDAS is extended to GEMDAS with G for Groupings.
Indeed, the long vinculum on the √ symbol is another example of that.
@@dazartingstall6680 How nice. When I was in school I had undiagnosed visual impairment. That fine a visual distinction would have been easily lost to me.
In other posts, when I tried to point out clarity is not a sin by using spaces to illustrate ambiguity, I was chastised for assuming spacing meant anything.
PEMDAS by any name is necessary. All I ask is that math be communicated clearly.
In another video in this channel, there was an expression something like 5/6x. PEMDAS says that's (5/6)x. To my eye, those parentheses don't add clutter, they derail misreading that as 5/(6x) - which it could have been if 6x had been under a horizontal line instead of a diagonal.
Parentheses can be overused. PEMDAS can be overused, too.
If the absolute number is just the positive of that number, why (ex. absolute number of -3 = +3 ) why do we even have a term for it. Why isn't it just written as a positive number in the first place. eliminating a need for the absolute symbol all together. I know I'm showing my ignorance with this question, but ignorance is curable, it's stupidity thar isn't.
We don't only do mathematics with numbers. Often we're working with generalised equations and formulae using algebra. And in that case, it can be useful to have a way to say "the magnitude of this variable" as distinct from its actual value.
A simple example of how this can be used is
√(x²) = |x|
@@gavindeane3670 Thank you, for answering my question. I genuinely appreciate you taking the time to explain why it is useful.
I know BODMAS
BUT FIRST TIME HEARING PEMDAS
BODMAS, PEMDAS, BIDMAS, BEDMAS, GEMDAS, PEMA, BIPS, BOPS, GEMS, etc...
There are lots of variations of the acronym. They all mean the same thing.
Too much yammering and not enough DOING!!!!
The idea is to teach those who don’t know not those who know … thus the talking . I got it quickly but I know it too
Sorry, but it is too long!
Thank you 💕
I find the teaching style includes so many chatty asides and repetitions that I lose focus on the main thrust of the lesson.
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