Question

which of the following is equivalent to \\(\sqrt5{32}\\) ?

a. \\(32^{\frac{1}{2}}\\)

b. \\(32^{5}\\)

c. \\(32^{\frac{1}{5}}\\)

d. \\(32^{\frac{1}{3}}\\)

Explanation:

Identify the given expression

We are given the radical expression:

$$ \sqrt[5]{32} $$

Apply the definition of rational exponents

The definition of a radical with index \(n\) written as a rational exponent is:

$$ \sqrt[n]{x} = x^{\frac{1}{n}} $$

Convert the expression

Using this definition with \(x = 32\) and \(n = 5\):

$$ \sqrt[5]{32} = 32^{\frac{1}{5}} $$

Compare with the given options

We match our result with the multiple-choice options:

• Option A: \(32^{\frac{1}{2}}\)
• Option B: \(32^5\)
• Option C: \(32^{\frac{1}{5}}\)
• Option D: \(32^{\frac{1}{3}}\)

The correct option is C.

Answer:

• (A) \(32^{\frac{1}{2}}\)
• (B) \(32^5\)
• (C) \(32^{\frac{1}{5}}\) (Correct answer)
• (D) \(32^{\frac{1}{3}}\)