Problem 108 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified expression for the given problem is \(x^{-8}y^{7}\)

Step 1: Expand using exponents rules

The given expression: \(\frac{(xy^{-2})^{-2}}{(x^{-2}y)^{-3}}\). Now we'll use power of power rule (a^{m^n} -> a^{m*n}), which states that to raise a power to a power, just multiply the exponents. Also remember that a^{-n} = 1/a^n. This expands to: \(\frac{x^{-2}*y^4}{x^6*y^{-3}}\)

Step 2: Separate terms

Separate the terms in fraction: \(x^{-2}/x^6 * y^4/y^{-3}\)

Step 3: Simplify terms using exponents rules

Using the rule which states that while dividing exponential terms with the same base, subtract the exponent at the bottom from the exponent at the top (a^m/a^n -> a^{m-n}), simplification leads to: \(x^{(-2-6)} * y^{(4-(-3))}\)

Step 4: Simplify further

Calculating the expressions in the brackets gives: \(x^{-8}y^{7}\)