Problem 114 - College Algebra Essentials

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\frac{\left(2^{-1} x^{-3} y^{-1}\right)^{-2}\left(2 x^{-6} y^{4}\right)^{-2}\left(9 x^{3} y^{-3}\right)^{0}}{\left(2 x^{-4} y^{-6}\right)^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(16 y^6 / x^{18}\)

Step 1: Simplify Negative Exponents

The first step is to simplify the negative exponents in the expression. A negative exponent translates to taking the reciprocal of the base. Thus, the expression:\n\( \left(2^{-1} x^{-3} y^{-1}\right)^{-2} \left(2 x^{-6} y^{4}\right)^{-2} \left(9 x^{3} y^{-3}\right)^{0} / \left(2 x^{-4} y^{-6}\right)^{2} \) \nbecomes \n\( \left({1/2} x^{3} y\right)^{-2} \left({1/2} x^{6} y^{-4}\right)^{-2} \left(9 x^{3} y^{-3}\right)^{0} / \left({1/2} x^{4} y^{6}\right)^{2} \)

Step 2: Simplify Expressions Raised to 0 power

Any number raised to the power of zero is 1. Thus, the expression above simplifies even further to become:\n \( \left({1/2} x^{3} y\right)^{-2} \left({1/2} x^{6} y^{-4}\right)^{-2} / \left({1/2} x^{4} y^{6}\right)^{2} \)

Step 3: Simplify Expressions Raised to Negative Power

The negative power translates to reciprocal of expression raised to the power. Therefore, the expression simplifies to: \n\(1/\left({1/2} x^{3} y\right)^2 1/\left({1/2} x^{6} y^{-4}\right)^2 / \left({1/2} x^{4} y^{6}\right)^2\) = \n\({4x^{-6} y^{-2}}{4x^{-12} y^8}/{4x^8 y^{12}}\)

Step 4: Simplify the expression

Now by simplifying the above expression we can cancel out the terms. We get: \n\( \frac{4x^{-6} y^{-2} * 4 x^{-12} y^8}{4x^8 y^{12}}\) = \n\(16 x^{-18} y^6\) = \n\(16 y^6 / x^{18}\)