Question

What is the cube root of w?
(1) The 5th root of w is 64.
(2) The 15th root of w is 4.

• A. Statement I alone is sufficient to answer the problem.
• B. Statement II alone is sufficient to answer the problem.
• C. Statement I and II both are needed.
• D. Either of the statements I or II is sufficient.
• E. Statement I and II both are not sufficient.

Answer 1

The correct option is D Either of the statements I or II is sufficient.

Arithmetic Operations with radical expressions

Determine the cube root of $w$, or determine the value of $w^{\frac{1}{3}}$.

(1) Given the 5th root of $w$ is 64, then $w^{\frac{1}{5}}=64$

and $(w^{\frac{1}{5}}{)}^{\frac{5}{3}}={64}^{\frac{5}{3}}$. Since $(w^{\frac{1}{5}}{)}^{\frac{5}{3}}=w^{\left(\frac{1}{5}\right)\left(\frac{5}{3}\right)}=w^{\frac{1}{3}}$, then $w^{\frac{1}{3}}=\left({64}^{\frac{5}{3}}\right)=({64}^{\frac{5}{3}}{)}^5=4^5$; SUFFICIENT.

(2) Given the 15th root of $w$ is 4, then $w^{\frac{1}{15}}=4$ and $(w^{\frac{1}{15}}{)}^5=4^5$. Since $(w^{1}15{)}^5=w^{\frac{5}{15}}=w^{\frac{1}{3}}$, then $w^{\frac{1}{3}}=4^5$; SUFFICIENT.

The correct answer is D; each statement alone is sufficient.