Problem 34 - College Algebra Essentials

Question

Simplify each exponential expression. $$\left(x^{-6}\right)^{4}$$

Step-by-Step Solution

Answer

The simplified form of \( \left(x^{-6}\right)^{4} \) is \( \frac{1}{x^{24}} \).

Step 1: Understanding the Problem

First, identify the base and the exponents in the expression. The base here is \(x\), the inner exponent is \(-6\) and the outer exponent is \(4\).

Step 2: Applying Exponent Rules

We apply the power of a power rule, which is \((a^{m})^{n} = a^{m*n}\). Substituting the base and the exponent values from our problem, we get \( (x^{-6})^{4} = x^{-6*4} \)

Step 3: Simplifying The Expression

Continuing from the previous step, we multiply \(-6\) by \(4\) to simplify the expression, thus getting \(x^{-24}\).

Step 4: Converting To Positive Exponent

A negative exponent can be converted to a positive exponent by taking the reciprocal of the base. Hence, \(x^{-24}\) can be rewritten as \(\frac{1}{x^{24}}\) which is a more convenient and standard form.