Problem 62 - College Algebra Essentials

Question

Simplify each exponential expression. $$\left(\frac{-30 a^{14} b^{8}}{10 a^{17} b^{-2}}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified exponential expression is \(-27 b^{30} / a^{9}\)

Step 1: Simplify the Numerical Part

Calculate the numerical expression first, without considering the variables. We see that -30 divided by 10 gives -3.

Step 2: Apply the Exponential Rules to the Variables

For variables with the same base, when you divide, you subtract the exponent in the denominator from the exponent in the numerator. Therefore, the exponent of \(a\) after the division is \(14 - 17 = -3\) and the exponent of \(b\) is \(8 - (-2) = 10\). So, the expression inside the parentheses after simplifying becomes \(-3 a^{-3} b^{10}\).

Step 3: Raise the Expression to the Third Power

According to the power of a power rule, when you raise an exponent to another exponent, you multiply the exponents. Therefore, after raising the simplified expression to the power of 3, the numerical part and each variable with their respective exponents become: \((-3)^3, (a^{-3})^{3}\) and \((b^{10})^{3}\). This simplifies to \(-27 a^{-9} b^{30}\).

Step 4: Simplify the Final Answer

In the final answer, negative exponents mean that the base is in the denominator of a fraction. Therefore, \(-27 a^{-9} b^{30}\) converts to \(-27 b^{30} / a^{9}\) as the simplified expression.