Sure! Let's work through the expression [tex]\((\sqrt[3]{a})^6\)[/tex] step by step.
### Step-by-Step Solution
1. Understand the given expression:
The expression we need to simplify is [tex]\((\sqrt[3]{a})^6\)[/tex].
2. Rewrite the expression using exponents:
[tex]\(\sqrt[3]{a}\)[/tex] can be rewritten as [tex]\(a^{1/3}\)[/tex]. So, we rewrite the given expression:
[tex]\[
(\sqrt[3]{a})^6 = (a^{1/3})^6.
\][/tex]
3. Simplify the expression using the exponentiation rule:
We use the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, [tex]\(m = \frac{1}{3}\)[/tex] and [tex]\(n = 6\)[/tex]. Therefore:
[tex]\[
(a^{1/3})^6 = a^{1/3 \cdot 6}.
\][/tex]
4. Simplify the exponent:
Multiply the exponents inside the power:
[tex]\[
1/3 \cdot 6 = 2.
\][/tex]
5. Write the final simplified expression:
Now that we have simplified the exponent, we get:
[tex]\[
a^{1/3 \cdot 6} = a^2.
\][/tex]
Therefore, the expression [tex]\((\sqrt[3]{a})^6\)[/tex] simplifies to [tex]\(a^2\)[/tex].
The correct answer is:
C. [tex]\(a^2\)[/tex]
