(The square root of 8 plus 1) divided by the square root of 8 =? Basic Algebra!
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- เผยแพร่เมื่อ 21 พ.ค. 2024
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![16 divided by [ (9 - 5) squared ] times 16 =? Can You Solve In Your Head?](http://i.ytimg.com/vi/MBlHXVxSuHA/mqdefault.jpg)
![16 divided by [ (9 – 5) squared ] times 16 =? Can You Solve In Your Head?](/img/tr.png)
You made this more complicated than necessary. An equally valid answer is 1 + 1/2*sqrt (2) and you can get there in 3 steps
(Sqrt(8) +1)/sqrt (8) = sqrt (8)/sqrt(8) + 1/sqrt(8)
Simplifying you get
1 + 1/sqrt(8) = 1 + 1/2sqrt(2)
Agreed, too much talking about how to do it wrong & never gets to answer. Never spend time on the wrong way, Always refer & APPLY the basic properties (Assoc, Com, Dist, Eq & introduce vinculum). Here numerator tell us how many denominators are added. The rest is confusion. KISS - they will figure out the rest.
I can't agree with you. What he is doing is showing not how quickly or in the least amount of steps you can solve it but logic behind all this and widen the knowledge how to get to the solution. This may be useful in more complicated math problems.
Too much bla, bla, bla.
exactly. ridiculous wading through this.
Answer starts at 8:10
Congratulations -- you made a simple process very complicated. ----> (4 + sqrt(2))/4
I can't agree with you. What he is doing is showing not how quickly or in the least amount of steps you can solve it but logic behind all this and widen the knowledge how to get to the solution. This may be useful in more complicated math problems.
@@user-ky5dy5hl4d Occam's razor. Best logic is the most efficient. Efficiency is important in mathematics, physics, and computer programming.
@@Dr_piFrog Correct. Logic is important in religions, too. Logic will lead to a conclusion that all religions are total BS.
If this is the correct answer, I would suggest the question should be: "Can you simplify to a common rational denominator?" I'm not sure how either the answer given or
1 + sqrt(2) / 4 "solves" the problem.
Well, it was (4 + √2)/4, but that is the simplest form. Not all simplifications give you something a LOT simpler. I think one point he was making here is that by convention, a radical should not be in the denominator. I don't really understand the reason for that (I've seen it in another video) but rules is rules, I guess.
Exactly. It's not a solution at all. Rationalising the denominator is not "simplifying" or "solving". Rationalising the denominator is just rationalising the denominator.
@@Astrobrant2(4+√2)/4 is no simpler than the original expression. Rationalising the denominator is not "simplifying" or "solving". Rationalising the denominator is just rationalising the denominator.
You say you don't really understand this convention of not having a radical in the denominator. That's probably because there is no good reason for such a convention. It's just needless dogma.
Indeed, in this very video he specifically uses a term that DOES have a radical in the denominator. He multiplies the entire thing by √8/√8, thereby demonstrating that radicals in denominators are not just OK, they can in fact be useful.
@@gavindeane3670u put me feelings into words. Thank you.
@@Astrobrant2 : The form with the "simpler" denominator would be easier to use if long division was the only method available for obtaining a decimal approximation.
Whilst it is true that (√8+1)/√8 is the same as (4+√2)/4, the idea that the first one is a question and the second one is the answer is just silly.
You can't have radicals in the denominator.
@@CAustin582 Of course you can have radicals in the denominator. It's perfectly fine. He uses a term with a radical in denominator in this very video. He multiplies the expression by √8/√8, thereby demonstrating that radicals in denominators are not just perfectly fine, they can also be useful.
Rationalising the denominator is not "solving" or "simplifying". Rationalising the denominator is rationalising the denominator. If he'd posed the question as "Can you rationalise the denominator?" then there would be no problem.
(√8+1)/√8 and (4+√2)/4 are just different but equally valid and correct ways of expressing the same value. This obsession he has with not permitting radicals in denominators is not mathematics. It's just needless dogma.
@@gavindeane3670 Of course it's fine to have radicals in the denominator in your steps or as part of the problem. The point is that it's not accepted as part of the solution.
@@CAustin582 It's not accepted as part of the solution BY THIS GUY. Mathematically it's perfectly acceptable and normal.
For example, the world is full of mathematicians and scientists and engineers who know that the sine and cosine of 45 degrees is 1/√2.
In this video he starts with an expression of the form (a+b)/c and he ends up with an expression of the form (a+b)/c. Regardless of what he says in the video, he has not in any way simplified the expression. It is exactly as simple as it was to start with. And "solve" is completely inappropriate for this.
@@gavindeane3670 Then you could also argue that 2/6 is just as valid of an answer as 1/3. Having a radical in the denominator doesn't break any rules of math; it's just a common convention for simplification. This guy definitely didn't make it up.
It doesn't require an 18 minute explanation !
Multiply top and bottom by sqrt(8). Get 8 as the denominator (rationalize 🤓), multiply out the nominator to 8 + sqrt(8), and go from there. Not yet watched the vid, but I feel a bit more can be done. However, for the purpose of the exercise, I guess, the denominator, I’d give at least 80% credit at this stage 😎
Here's an idea. How about getting right to solving (simplifying) the problem instead of showing us 5 ways how NOT to solve it? 95% of those who will take the time to watch the solution akready understand the concepts of identity and distribution, so it's not necessary to show us. For the few who don't , refer them to a separate tutorial about those things, and when they are good with that, THEN they can come back and apply those concepts to this problem. 90% of this video is a waste of time for 95% of the viewers.
I don't believe that this site is for people who want to proclaim their mathematical genius to the world, but instead is for dullards like me.
This can be simplified in five lines in less than 10 seconds.
1 + (1/sqrt(8))
=1+ sqrt(8)/8
oops
1 +(sqrt(8)/8)
No. It equals (8 + √8)/8. You forgot to multiply _both_ of the terms in the numerator by √8.
Also, you used the parentheses wrong in your first line. It should be (√8 + 1)/√8
No one thinks the answer is 1. This was way more confusing than it needed to be.
There isn't a solution to an expression. You might simply it if clearly stated rules describing simplest forms are given first.
I would have separated the numerator into two terms rationalizing the second term and left the answer as 1+ (sqrt(2)/4).
A really great explanation. I will take your Math Skills Rebuilder and your Pre Calculus courses just for the pleasure of learning.
(4+ sqrt 2)/4 is not finished.
By partial fraction decomposition it becomes 1 + (1/(2* sqrt 2)).
Like this.
Sqrt 8= 2 * sqrt 2.
((2 * sqrt 2)+1)/(2* sqrt 2)
Decompose, the 2 sqrt 2's cancel,
Leaving (1 + (1/(2*sqrt 2))
so, what is the answer ? (mine is 5/4sqrt2 )
Thank you. I home school my son, so I love to keep up with my Math Skills. I keep telling him that math in our language is known as " Mathafu" which deriveds from the word "Ma" meaning truth. This is commom in Bantu. It all comes from ancient Kemet, when the Atlantians, Tehuti and the rest of them arrived to teach the Kemites the truths and universal laws of the universe.
And your point is ?????
@@kingalfred3902 "Ma" what you call math is occult. If there is no limit to infinity ♾️ then zero is also an illusion. Tahiti said, "that which has a beginning has an end. And that which has no beginning has no end." We have totally misunderstood "zero" , that's why your calculator cannot divide by zero. Example: 1÷0 Once you understand this then, you are now knee deep in the unexplainable realm, which you call "Black Magic" or "The God Realm" where the laws of science do not apply.
Thank you
Solve or simplify? The genius at work again.
I guess I don't understand how the "answer" is really any better than the "problem".
You want to get an integer plus or minus a fraction with whole number in the denominator to get a standard value.
@@kimobrien.You'll need to try that again. That didn't make any sense.
The person you replied to raised a completely valid point. The "answer" in this video is not a solution to or simplification of the original expression. It's just a different way of expressing the same thing.
@@gavindeane3670 You do this type of simplification in preparation for a decimal approximation. Doing the approximation of any root and dividing by a whole number is almost always much simpler than dividing a whole number by the decimal approximation or doing multiply decimal approximations and then multiplication or division. Also the smaller the number under the root sign the easier it is to calculate the approximation. It also provided a standard way to compare numbers like this. When I was in HS we didn't have hand held calculators either.
@@gavindeane3670 The correct answer is 1+√2/4
@@kimobrien. Rationalising the denominator is not simplifying. Rationalising the denominator is rationalising the denominator. It might be a useful thing to do sometimes, but he isn't presenting us with a particular requirement of some operation or calculation we need to do next. You are just assuming that the thing we need to do next is a thing that would be easier without a radical in the denominator. Even if that were true it's still not correct to describe the re-expression of (1+√8)/√8 as (4+√2)/4 or 1+√2/4 as "simplification". It's just expressing the same thing in a different way.
In the video he claims we are "solving" or "simplifying" the original expression. That's nonsense. All he needs to do is change the question to "Can you rationalise the denominator?". Then (4+√2)/4 and 1+√2/4 would be correct answers to that question. As the question is currently posed, (4+√2)/4 and 1+√2/4 cannot reasonably be described as "answers" at all.
Thank you. I was able to get the right answer.
YEA SURE ...!!!!.....
Yeah, me too, I just read down the comments to find the answer.
Could you make it any harder.
😂
Or if you separated the two terms, you could have 1+((1/4)sqrt(2)).
super simple:
First step: reduce √8 + 1 to 2√2 + 1. Numerator is now 2√2 + 1
Second step: denominator is irrational due to the root sign so we rationalize it by multiplying the numerator and the denominator by √8
Numerator: (2√2 + 1) * √8 which is 2√16 + 2√2 which reduces further to 8 + 2√2.
Now the denominator: √8 * √8 is just 8.
Third Step reduce fraction further. 8, 2 , 8 can all be divided by 2 so factor out a 2 and we get the final answer of (4 +√2) / 4.
Because BOTH the 4 AND the √2 are divided by 4 we can NOT reduce this further.
Like I said super simple if you know what to do it should take no more than 30 seconds to do. (This was a step by step explanation so it makes it look like more work than it actually is)
Solve? Is it equation? I don't think so.
3/2
thanks for the fun.
Thanks. Good video. I'm still smart at math even though I'm stupid at everything else in life.
I'll bet you're not!
@@nickfraser4599 muted
What I'd like to know is how the answer is the problem simplified. All that work and still the answer is not simplified.
It's not any simpler, is it.
What he's actually done is rationalised the denominator, not simplified.
Seeing problems and methods like this makes me wonder how I ever made it through algebra. There must be some other way.
Good explanation for beginners. The truth is math often gets reduced to a series of tricks to squeeze out the answer. If you don't know the tricks it is difficult to intuitively find the answers. Yes, math people call them rules but essentially they are the tricks to solving equations. Students looking for a more straight forward process are often confused when confronted with these situations. Compounding the issue is there may be multiple ways of finding alternate forms of the answers. Often successful math skills come down to "training" the person to answer a problem in a certain way similar to Pavlov teaching his dogs. Not kind but often true.
So true. I am having problems with Algebra because it seems so arbitrary and impractical.
How about 1 + (square root of 2 ) divided by 4 ???
3 over 2
When you say, "Can you solve?," that means, "What is the numerical value of this expression?" In this case, it is 1.3536. The question should be, "Can you reduce this expression?" For some reason, the presenter has an obsessive attachment to the idea that no expression should have a radical in the denominator. It's almost as bad as his near obsession with PEMDAS. In the Real World, where we want to see numbers, it makes absolutely no difference if there is a radical in the denominator or not.
🤓☝️
It's not even "Can you reduce this expression?". It's just "Can you rationalise the denominator?". That's literally all he's done.
The expression starts in the form (a+b)/c and by the end it's still in the form (a+b)/c.
Silver hammer. You are perfectly correct. Can you solve what?. You really hammered him. Good work.
John is just wasting our time. He should visit the John with sandpaper
@@harrymatabal8448 Rough crowd in here today...😁😲
(4+sqrt2)/4
Gotcha!
Basically 1 round up.
1.28868(approx.)
You can't solve this problem but you can simplify by rationalizing the denominator.
Wouldn't 1 + √(2)/4 be simpler? Seems weird to include the 4/4
The distributive property is taught in third grade. You made a simple problem very complicated.
Verdade
1.5 rounded to tenths
Multiply top and bottom by SR8 = (1 +SR8)/8. I failed to simplify it further by realizing that SR8 = 2*SR2
(√8+1)/√8 equals √8/√8+1/√8 equals 1+1/√8. Why is that not considered fully simplified?
1+(1/(2 sqrt 2))
Irrationals in the denominator are frowned upon, because they are way harder to calculate.
Also sqrt(8) can be simplified to 2sqrt(2).
So 1+1/sqrt(8)= 1+1/(2sqrt(2)) = 1+sqrt(2)/4
This guy is saying "simplify" when what he really means is "rationalise the denominator".
What he's doing here isn't simplification at all. His end point (and your end point) aren't any simpler than the starting point.
1/rut 2
The square root of 16 is 4. The square root of 8 is @2.89. One-half of 8 is 4.
Good god. I saw the problem and thought “how can this equation be simplified down to a rational number?” 18 minutes I will never get back. I recommend you teach civics or something.
All you get is another formula. What is that in practical terms? Would it not be 1.353....?
1.0607
Oh man I got 1+sqrt2/4 I guess I better keep studying.
Your answer is better!!!
How about 1/4 * (4 + sum_(k=0)^∞ ((-1)^k (-1/2)_k (2 )^k )/(k!) ) No square root at all, just an infinite sum. Basic limits! Next challenge is to write it as an integral, no square root allowed.
1+sqr root of 8
Quanta enrolação para resolver um problema elementar. Fala sem parar
Simplify, not solve (in the title) please.
"Rationalise the denominator" not "simplify" (and yes, definitely not "solve").
Mathematicians are always “rationalizing” their beliefs. Lol😊
Can’t have the Square Root in the denominator and the answer is definitely not 1 May I suggest multiplying the top and the bottom by the SR of 8
Square roots in denominators are fine. They can even be useful - as you (and the video) demonstrate when you multiply this by √8/√8
One simple question, Given the number 4:
The square roots of 4 are +2 and -2. That needs to accounted for!
It's doesn't need to be accounted for. The √ symbol means "principal square root of", which is the positive square root.
Yes, 4 has two square roots, but √4 refers only to one of them. √4 is 2.
If you want to refer to both square roots you put ± in front of the √ symbol.
nearly 1.166
1+V2/4.
break it down even further 1 + 1/4 x sqrt 2
I tried, failed - will watch video now
How is sq root of 8, 64? 8 squared is 64.
2
my first answer would be 1+1/sqrt(8)
What about 1 + (1/sqrt(8))?
Simplify sqrt 8 to 2 * sqrt 2
1+ 1/(2 sqrt 2)
That's also fine. You could express the √8 bit as 2√2 if you wanted to as well.
-1
Why wouldn't you make it
1 + (√2/4)?
Doesn't the distributive property work in subtraction too?
(4+√2)/4 would become 4/4 + √2/4, which would reduce to 1 + √2/4.
You could look at it as distributing 1/4 times the components of the numerator.
You absolutely can express it as 1+√2/4. Nothing wrong with that at all.
It doesn't make sense to describe either 1+√2/4 or (4+√2)/4 as a "solution" to the original expression though. They're all just equivalent ways of saying the same thing.
I got that also. I broke the problem into 2 fractions which resulted in the 1 (like in your answer) with a denominator of 1 also. John's answer just combined both over a single denominator (4).
4 steps,
Ans: 1 + (sqr(2))/4
wrong
1
Ajab khan khattak.
Deviation...........deviation to
arrive at evaluation.
It seems very long😂
Chat GPT4 got 1+√2/4
That's the same as what he gets in the video, just expressed in a slightly different way.
The video and the comments show the precarious state of mathematics in 2024. The problem does not look for a solution but shall require a simplification, or a rational denominator. The vast majority of comments don't understand the problem either.
All he needed to do was change the question from "Can you solve?" to "Can you rationalise the denominator?"
Huinya
So what is the answer!!! 😂
That isn't algebra. There is no unknown.
Why wouldn’t the 8s cancel ? Not the 8 root , but the simplified ones
B/c the numerator is separated by a plus sign; you can’t cancel when the numerator is separated by a plus sign
"Multiplied by"not "times" FFS
Solve??? No one can solve this because it is simply an expression, not something that can be solved such as an equation. Really shocking and disappointing that a math teacher doesn’t know the difference between solving and simplifying. Typical sloppy, irresponsible wording by someone who cares only about “likes” and “views” and has little or no interest in imparting knowledge.
Yeah, I noticed that, too. He's done it in other videos, as well. While the proper instruction would be "simplify", I can't really get bent out of shape over using the word, "solve". But then, you and I just see things differently, and that's okay.
@@Astrobrant2The proper instruction here wouldn't be "simplify". The proper instruction would be "rationalise the denominator".
@@gavindeane3670 Why tell them that? I mean, if the teacher wants to give them a clue, then okay, I won't object to that. But if rationalizing the denominator is something the students are expected to know and are being tested on, then "simplify" seems appropriate.
@@Astrobrant2Because rationalising the denominator is not simplifying. The "answer" in this video is not in any meaningful way a simplification of the original expression. They are just different ways of expressing the same thing.
If students are expected to be able to rationalise a denominator then that's fine. Test them on that. But ask them to do what you want them to do. If the requirement is to rationalise the denominator then the instruction should be "rationalise the denominator". It is absurd and inexcusable to obfuscate that requirement behind a different instruction that literally does not describe what you are asking them to do.
2 roots cancel each other out and your left with 1
No, they do not cancel. He explained that. I'll give you the short version. You can only cancel like that if the two numbers in the numerator are being _multiplied or divided,_ not added or subtracted.
Nope
You can only 'cancel' like that if you are multiplying or dividing .. not adding or subtracting.
For example (A x B) / A = A but (A + B) / A IS NOT = A
@@MrMousley It was late and i forgot about how to use surds and the root function - got ahead of myself. Should of known better.🤣
@@robertloveless4938 Forgot how to use surds it was late and got ahead of myself i feel silly lol.
Watching you solve this problem was like watching water boil at Leadville Colorado on a very cold day.
(SQRT(8) + 1)/SQRT(8) = (2*SQRT(2) + 1)/(2*SQRT(2) = 3.8285/2.8285= 1 + 1/2.8285 =
1/1.4142 = .7070 this divided by 2 is .354 so 1.354
SQRT(2) * (2 * SQRT(2) + 1)
---------------
SQRT(2) * 2 SQRT(2)
( 4 + SQRT(2) )/ 4 = 1 + SQRT(2)/4 = 1.354
No calculators
Another way
1 + SQRT(2)/4 = 2.5 * SQRT (2)/10 + 1 = (2.8285 + .7070)/10 + 1 = 3.5354/10 + 1 = 1.35354
SQRT(8) + 1/ SQRT(8) = 1+ 1/SQRT(8) * SQRT(2)/SQRT(2) = 1 + SQRT(2)/SQRT(16) = 1 + SQRT(2)/4
1 + 1/SQRT(8) = 1 + 1/2 *1/SQRT(2) = 1 + .7070/2 = 1.354
You are excelent but you talk too much😊
sorry but there got to be a better way to explain this problem! I get it that the instructor knows how to but his job is to make math EZ so we don't have to use calculators and computers for simple problems!
here is an exact quote of my comment above, let me know if you find it easier to understand:
"super simple:
First step: reduce √8 + 1 to 2√2 + 1. Numerator is now 2√2 + 1
Second step: denominator is irrational due to the root sign so we rationalize it by multiplying the numerator and the denominator by √8
Numerator: (2√2 + 1) * √8 which is 2√16 + 2√2 which reduces further to 8 + 2√2.
Now the denominator: √8 * √8 is just 8.
Third Step reduce fraction further. 8, 2 , 8 can all be divided by 2 so factor out a 2 and we get the final answer of (4 +√2) / 4.
Because BOTH the 4 AND the √2 are divided by 4 we can NOT reduce this further.
Like I said super simple if you know what to do it should take no more than 30 seconds to do. (This was a step by step explanation so it makes it look like more work than it actually is)"
Hope you found that easier.
By this video, you clearly explained why nobody should take your lessons. I am fortunate to have had much better teachers than you are.
Not done yet . End answer is 1 + (1/4)sqrt(2)
Annoyingly unnecessarily long winded!!
The written question should be "Can you simplify?"
There are two simplified answers; the one shown and 1+√(2)÷4.
It's not even "Can you simplify?". It's "Can you rationalise the denominator?". That's all he's done.
@@gavindeane3670 You rationalize to simplify the expression.
@@sandikay3323 Rationalising the denominator is not simplifying. Rationalising the denominator is rationalising the denominator.
He starts with an expression of the form (a+b)/c and he ends up with a different expression of the form (a+b)/c. It's nonsense for him to suggest the end point is any simpler than the starting point.
There's no sensible justification for the notion that (√8+1)/√8 is a question and (4+√2)/4 is the answer, let alone considering one to be a simplification of the other.
This is a joke right ?
😂
Still not clear ? Neds better explanation even from clowns in this conment section!
The KJV Bible is mathematically encoded
Theomatics by Jerry Lucas. VERY INTERESTING.
One plus the √2
Simplify, not solve...
It's not simplify either.
What he's doing is rationalising the denominator.
@@gavindeane3670 Yes, he is rationalizing the denominator, which is a form of "simplify," since it is not correct to leave a fraction with any kind of root in the denominator... :)
@@cmills14916 Rationalising the denominator is not simplifying. Rationalising the denominator is Rationalising the denominator.
There is nothing incorrect about roots in denominators. That's just silly dogma that some people are subjected to in schools. In the real world nobody cares. It is completely normal. For example, engineers and scientists and mathematicians the world over know that the sine and cosine of 45 degrees is 1/√2.
The notion that the original expression in this video is in some way a problem, and the expression he ends up with is in some way an answer to that problem, is just daft.
So, I'm daft...👍🏼
you lost me.
You are not finished yet
1 + 1/4 * Sqrt(2)
That's the same as what he ended up with, just expressed slightly differently.
Too much talk!
1+2√2/8 becomes 1+√2/4 That is the correct answer for simplification. NOT (4+√2)/4
You are very bad math teacher your explanation is bad this problem could solve a lot easier and simpler
Too long, too much bla bla I nearly fall asleep.
I started this and paused it just to leave a comment. The thumbnail asks "Can You Solve?" Solve what? Solving implies there is an equation for which the solutions are sought. This is not an equation; there is no "=" sign anywhere. It is just an algebraic expression. The thumbnail should have asked either to rationalize the denominator or simplify. Furthermore it is a relatively trivial problem that does not require an 18 minute video (even for viewers not very good at math). I'm guessing you dragged out the video so long just to get more money from ads.