Problem 107 - College Algebra Essentials

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\frac{\left(x^{-2} y\right)^{-3}}{\left(x^{2} y^{-1}\right)^{3}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(x^{-12}\)

Step 1: Apply the Power Rule (a^(m*n) = a^(m)*a^(n))

The expression can be rewritten as: \((x^(-2*3) y^(-3)) / (x^(2*3) y^(-1*3))\). Therefore the expression becomes: \((x^{-6}y^{-3}) / (x^{6} y^{-3})\)

Step 2: Use Law of Exponents to Combine Like Terms (a^m / a^n = a^(m-n))

This simplifies the expression to: \(x^{(-6-6)} y^{(-3-(-3))}\). Then the expression becomes: \(x^{-12} y^0\)

Step 3: Simplify the Expression

Any nonzero number to the power of zero is 1. Hence: \(y^0 = 1\). The final simplification of the expression is: \(x^{-12}*1\)