Multiply Cube Roots: Solving ∛5 × ∛5

Solve the following exercise:

$\sqrt[3]{5}\cdot\sqrt[3]{5}=$

In order to simplify the given expression, we will use two laws of exponents:

a. Root definition as an exponent:

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

b. Law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's start by converting the roots to exponents using the law mentioned in a':

$\sqrt[\textcolor{red}{3}]{5}\cdot\sqrt[\textcolor{red}{3}]{5}= \\ \downarrow\\ 5^{\frac{1}{\textcolor{red}{3}}}\cdot5^{\frac{1}{\textcolor{red}{3}}}=$

We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b':

$5^{\frac{1}{3}}\cdot5^{\frac{1}{3}}= \\ 5^{\frac{1}{3}+\frac{1}{3}}=\\ 5^{\frac{1+1}{3}}=\\ \boxed{5^{\frac{2}{3}}}$

Therefore, the correct answer is answer b'.