Multiply Cube Roots: Solving ∛(2²) × ∛2 Step by Step

To simplify the given expression we use two laws of exponents:

A. The law of roots (expanded):

$\sqrt[\textcolor{blue}{n}]{a^{\textcolor{red}{m}}}=a^{\frac{\textcolor{red}{m}}{\textcolor{blue}{n}}} =(\sqrt[\textcolor{blue}{n}]{a})^{\textcolor{red}{m}}$

$$B. The law of exponents for multiplication of terms with identical bases:

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

Let's start from the root level to write exponents using the law of exponents shown in A:

$\sqrt[\textcolor{blue}{3}]{2^{\textcolor{red}{2}}}\cdot\sqrt[\textcolor{blue}{3}]{2}= \\ \sqrt[\textcolor{blue}{3}]{2^{\textcolor{red}{2}}}\cdot\sqrt[\textcolor{blue}{3}]{2^{\textcolor{red}{1}}}= \\ \downarrow\\ 2^{\frac{\textcolor{red}{2}}{\textcolor{blue}{3}}}\cdot2^{\frac{\textcolor{red}{1}}{\textcolor{blue}{3}}} =$

We continue, since multiplication is performed between two terms with identical bases - we use the law of exponents shown in B:

$2^{\frac{2}{3}}\cdot2^{\frac{1}{3}}= \\ 2^{\frac{2}{3}+\frac{1}{3}}=$

We continue and perform (separately) the operation of combining the numerators in the exponent fraction that was obtained, this is done by expanding each of the numerators to the common denominator - the number 3, then we perform the addition and subtraction operations in the numerator of the fraction:

$\frac{2}{3}+\frac{1}{3}=\\ \frac{2+1}{3}=\\ \frac{3}{3}=\\ 1$

In other words - we get that:

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

Let's summarize the process of simplifying the expression:

$\sqrt[3]{2^2}\cdot\sqrt[3]{2}= \\ \downarrow\\ 2^{\frac{2}{3}+\frac{1}{3}}=\\ \boxed{2}$

Therefore, the correct answer is answer A.