Problem 41 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(-64/x^3\).

Step 1: Analyze the expression

In the given notation \(\left(-\frac{4}{x}\right)^{3}\), the main operation is the exponentiation. The exponential operation is denoted with the symbol ^, and the number 3 is the exponent. In this exercise, the exponent is a positive integer, so we simply need to replicate \(-\frac{4}{x}\) three times and multiply them.

Step 2: Apply exponentiation

Since the exponent is 3, which is an odd number, the negative sign remains due to the property of negative numbers raised to odd powers. We replicate \(-\frac{4}{x}\) three times, multiplying them together: \(-\frac{4}{x} * -\frac{4}{x} * -\frac{4}{x}\).

Step 3: Simplify the expression

We simplify the expression: \(-4*-4*-4 = -64\) and \(\frac{1}{x} * \frac{1}{x} * \frac{1}{x} = \frac{1}{x^3}\), so the simplified expression is \(-64/x^3\).