Problem 111 - College Algebra Essentials

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(\frac{x^{3} y^{4} z^{5}}{x^{-3} y^{-4} z^{-5}}\right)^{-2}$$

Step-by-Step Solution

Answer

The simplified expression is \(1/(x^{12} y^{16} z^{20})\).

Step 1: Deal With Negative Exponents

First, apply the rule that \(x^{-n} = 1 / x^n\). Use this property to switch the positions of variables in the denominator with negative powers, to the numerator and vice versa, while changing their power to positive: \(\left(\frac{x^{3} y^{4} z^{5}}{x^{-3} y^{-4} z^{-5}}\right)^{-2} = \left(\frac{x^{3+3} y^{4+4} z^{5+5}}{\underline{\phantom{xx}}}\right)^{-2}\).

Step 2: Simplify the Exponents

Add up the exponents of like bases. This results in: \( \left(x^{6} y^{8} z^{10}\right)^{-2}\).

Step 3: Use the rule \( (x^m)^n = x^{m \cdot n} \)

Use the power of a power property, which states that when you have an exponent raised to another exponent, you multiply the exponents. Applying this rule will result in: \( x^{6 \cdot -2} y^{8 \cdot -2} z^{10 \cdot -2}\).

Step 4: Calculate the final expression

Do the multiplication inside the exponents. This will result in: \( x^{-12} y^{-16} z^{-20}\). It's clean to present the final expression without any negative exponents. So, applying the rule \( x^{-n} = 1/x^{n} \) will result in: \( 1/x^{12} * 1/y^{16} * 1/z^{20} = 1/(x^{12} y^{16} z^{20})\).