Problem 61 - College Algebra Essentials

Question

Simplify each exponential expression. $$\left(\frac{-15 a^{4} b^{2}}{5 a^{10} b^{-3}}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified expression is \( -27 a^{-18} b^{15} \)

Step 1: Rewrite the Expression Using Exponent Rule

The expression inside the parenthesis is a division. According to the division rule of exponents, we subtract the powers when bases are the same and the terms are divided. So, rewrite the expression as: \( \left(-3 a^{-6} b^{5}\right)^{3} \)

Step 2: Apply Power of a Power Rule

Next, we apply the power of a power rule which means you multiply the powers when there's exponentiation of powers. Doing this, the expression is rewritten as: \( -3^{3} a^{-18} b^{15} \)

Step 3: Simplify the Coefficient

Finally, we simplify \( -3^{3} \) which results in -27. So, the simplified exponential expression is: \( -27 a^{-18} b^{15} \)