Problem 55 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified form of \( (4x^{3})^{-2} \) is \( \frac{1}{16x^{6}} \).

Step 1: Apply negative exponent rule

The negative exponent indicates a reciprocal. Therefore, rewrite \( (4x^{3})^{-2} \) as \( \frac{1}{(4x^{3})^{2}} \).

Step 2: Apply power of a power rule

The power of a power rule states that when raising a power to a power, the exponents should be multiplied. Therefore, you multiply the exponent '3' of \( x \) by the outer exponent '2' to get \( \frac{1}{4^{2}x^{3 \cdot 2}} \).

Step 3: Simplify result

Now calculate and simplify your expression. \(\frac{1}{4^{2}}\) becomes \(\frac{1}{16}\). \(x^{3 \cdot 2}\) becomes \(x^{6}\). Therefore, the simplified expression is \( \frac{1}{16x^{6}} \).