A question that CONFUSED many teachers
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- เผยแพร่เมื่อ 12 ก.ย. 2024
- What is a correct answer? If you're reading this ❤️.
A great math questions:
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All confusion comes from the reading of the problem. (16^16)^16 =/= 16^(16^16). The square root is not the stumbling block.
Yes, if we don't remember the precedence rules, then we can get the wrong values. When writing in a linear line of text, it's best to use extra pairs of parentheses to avoid confusion. Good software tools evaluate left to right for + - * / and give higher precedence to * / over + -, unless the latter are unary operators. However, exponentiation has higher precedence over * /, and successive exponentiation has right to left precedence.
Nice problem but do it all as powers of 2 from the start seems a bit easier to me.
Eg sqrt (2^4)^(2^64)
Which is sqrt (2^ (2^66))
2^(2^65)
yooo i did the same thing 😀
16^2^63 is "ugly" => 4^16^16
x = (4*4)^(16^16) = 4^16^16 * 4^16^16, so √x = 4^16^16
Or: (4^2)^(16^16) = 4^(2*(16^16)) = [4^(16^16)]^2 => 4^(16^16)
But this doesn't look like any of the choices, so you still need to manipulate the numbers until you find a match, or start evaluating the choices.
@@oahuhawaii2141 Yes, of course you're right. Sadly it won't help with the choices. It just is easier and more beautiful than the right choice. But it may help you some other time. Thank you for taking your time to analyse.
Great step-by-step solution!
√16^(16^16)=16^(2^-1 * (2^4)^16)=16^(2^-1 * 2^64)=16^(2^63)
why can't you treat it as the sqrt((16^16)^16) then use the division you showed earlier to make it (16^16)^(16/2) which is 16^16^8?
It's not ((16^16)^16).... It's (16^(16^16)) Where the answer will change completely......... You just can't swap brackets like that. The meaning of the question will change
The precedence rule on successive exponentiations is right to left: a^b^c^d means a^(b^(c^d)). Any problem that requires left to right order will use parentheses, as in ((a^b)^c)^d. Thus, (a^b)^c^d means (a^b)^(c^d) .
Why not simplify it one step further: 2^(2^65)
Because none of the choices of answers are in that form. It's possible to reduce each choice to the form of 2^(2ⁿ), but that's extra work. I did it for fun.
@@oahuhawaii2141 But you should aways do that with silly multiple-choice questions! (Showing that the choices offered are incomplete or sub-optimal, I mean...)
To evaluate √(16^(16¹⁶)), we can take the square root of the base, 16, to get 4, or take half of the exponent, 16¹⁶, to get 16¹⁶/2. In the first case, we have 4^(16¹⁶), which mismatches case a.), so we reject it. The other 3 cases retain the base of 16, so we look at how to evaluate the exponent of 16¹⁶/2 for a match. We can toss out case c.), as halving 16¹⁶ doesn't halve the exponent of 16 to 8. In case b.), the exponent of 4⁴ can be recast as 16², which is far from 16¹⁶/2. So, we evaluate 16¹⁶/2 to see that it is (2⁴)¹⁶/2¹, which is 2^(4*16-1), or 2⁶³. This matches d.) 16^(2⁶³) .
Alternatively, we can evaluate everything into powers of 2:
√(16^16¹⁶) = 4^((2⁴)¹⁶) = (2²)^(2⁶⁴) = 2^(2*2⁶⁴) = 2^(2⁶⁵)
a.) 4^(4⁴) = 4^(2⁸) = 2^(2 * 2⁸) = 2^(2⁹)
b.) 16^(4⁴) = 16^(2⁸) = 2^(4 * 2⁸) = 2^(2¹⁰)
c.) 16^(16⁸) = 16^(2³²) = 2^(4 * 2³²) = 2^(2³⁴)
d.) 16^(2⁶³) = 2^(4 * 2⁶³) = 2^(2⁶⁵) . This is a match.
If you know powers of 2 and realize you can rewrite the 16 as 2^4 it's pretty straightforward that once you cancel the sqrt you end up with D.
The 16 appears in 3 places. You must choose the right ones to replace.
I originally went to (C) but later realized I had missed the implicit grouping of the radical sign. sqrt(16^16^16) is (16^16^16)^(1/2), not (16^16^16^(1/2)) or some other weirdness.
I saw the problem as:
sqrt(((16)^16)^16)=((16)^16)^8
So the answer is C.
👍 right from minute 1:27
Power towers are always evaluated from the top down, so it's implicitly it's 16^(16^16). Look up tetration on Wikipedia. Alternative notations are ³16 and 16↑↑3.
@@nowster Thanks. I am in my 70s, so it's been a long time since I learned this concept. I will try it the correct way.
d
16^2^63
I haven't watched it yet, but my answer is C. I see that the only other answers here so far are for D. Well, maybe I will see where I went wrong after I watch the video.
The precedence rule on successive exponentiations is right to left: a^b^c^d means a^(b^(c^d)). Any problem that requires left to right order will use parentheses, as in ((a^b)^c)^d. Thus, (a^b)^c^d means (a^b)^(c^d) .
This man ia talking like a used car salesnan
Ordinary people need to think as the solution is dispensed but how can they think when he speaks faster than the speed of light
You can change the playback speed, and hit pause/play and rewind as needed.
@@oahuhawaii2141 or buy software to reduce speed