Simplify each expression.
${\left(4x^2\right)}^3$

• A. $4x^6$
• B. $12x^5$
• C. $64x^6$
• D. $64x^5$

Answer: (C)

Verified step by step guidance

1. Recall the Power of a Product Rule, which states that for any numbers \(a\) and \(b\), and any exponent \(n\), the expression \((ab)^n\) can be rewritten as \(a^n b^n\).
2. Identify the base terms inside the parentheses and the exponent outside the parentheses in the given expression \((ab)^n\).
3. Apply the Power of a Product Rule by raising each factor inside the parentheses to the exponent \(n\), rewriting \((ab)^n\) as \(a^n b^n\).
4. If the problem includes numerical values or variables, express each term separately with the exponent applied, for example, if \((2x)^3\), rewrite as \(2^3 x^3\).
5. Simplify each term if possible, such as calculating \(2^3\) to get \(8\), but do not combine terms with different bases unless specified.