What is the cube root of w?

What is the cube root of w?

(1) The 5th root of w is 64 --> \(\sqrt[5]{w}=64\) --> we can find \(w\), hence we can find \(\sqrt[3]{w}\): \(w=64^5\) --> \(\sqrt[3]{w}=\sqrt[3]{64^5}\). Sufficient.

(2) The 15th root of w is 4 --> \(\sqrt[15]{w}=4\). The same here. Sufficient.

Answer: D.
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1. If the nth root is asked for a number X:
If n is odd - It has a unique solution that satisfies.
On the other hand, if n is even - then both (y)^n and (-y)^n satisfies the solution.
Now to the problem:
A - sufficient. 5th root - 5 odd.
B - sufficient. - 15 -odd.

Answer D.

This is a simple Surd problem i will rate its difficulty at 500 or below.
as the first step break all numbers down to their basic form 64 = 4^3 = 2^6 ( we will find later that the second part is not necessary in this question, but in some other question we might need to break down further)
Lets take statement 1 : (w)^1/3 = 64 this means w = 64^3 = 4^15 = 2^30 ( stop ! much before this coz you knwo you can find it )
Second statement (w)^1/15 = 4 This means w = 4^15 = 2^30 ( Stop as soon as you know it is solvable)
So choice D is the nswer.

We need to determine the cube root of w. Thus, if we have a value for w, we can determine the value of the cube root of w.

Statement One Alone:

The 5th root of w is 64.

Recall that the nth root of a number is the number raised to the 1/n power; we can set up an equation with the information from statement one.

w^(1/5) = 64

Now raise both sides of the equation to the 5th power.

w = 64^5

Since we know that we have a unique value for w, we can stop here. This is enough information to enable us to determine the value of the cube root of w. Statement one provides enough information to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

The 15th root of w is 4.

We can set up an equation with the information from statement two.

w^(1/15) = 4

w = 4^15

Since we know that we have a unique value for w, we can stop here. This is enough information to enable us to determine the value of the cube root of w. Statement two is also sufficient.

The answer is D.
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Hi guys

what if we are given that 6th root of W is 64. In that case can we say w= 64^(6)? or will we say that modulus W = 64^(6). The clarification is more towards what if we are given even number instead of an odd number than can we concretely find the value of w by multiplying with an exponential power on both sides.

If \(\sqrt[6]{w}=64\), then \(w = 64^6\) only. w cannot be negative (w cannot be -64^6) because even roots from negative numbers are not defined for the GMAT.
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Hi

If the question talks about roots (not powers) then the answer will be only one. So if it says:
6th root of W = 64, then W = 64^6

We cannot say |W| = 64^6, because then we have to take a case where W = - 64^6, but then 6th root (or any even root) of a negative number is not defined.

If instead the question says: 6th power of W is 64, here we are given W^6 = 64 or W^6 = 2^6
Here definitely there will be two cases: W can be = 2 or W can be = -2; because both 2^6 and (-2)^6 give the same result, 64.

pushpitkc how do we call this process mathematically w = 64^3 = 4^15 = 2^30 exponential expansion or

And how from this 64^3 we get this 4^15 ?

many many thanks

\(64^3 = (2^6)^3 = 2^{6*3} = 2^{18}\)
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Target question: What is the cube root of w?

Statement 1: The 5th root of w is 64.
ASIDE: If the square root of x = 7, then x = 7²
If the cube root of x = 10, then x = 10³
If the fourth root of x = 6, then x = 6⁴
etc
So, if the 5th root of w is 64, then w = 64⁵
Since we COULD determine the actual value of w, we COULD find the cube root of w
In other words, we have all of the information we need to answer the target question with certainty
Statement 1 is SUFFICIENT

Statement 2: The 15th root of w is 4
From this we can conclude that w = 4¹⁵
Since we COULD determine the actual value of w, we COULD find the cube root of w
In other words, we have all of the information we need to answer the target question with certainty
Statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

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