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Chapter 0: Problem 111

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

Short Answer

Expert verified
The simplified form of the expression is \(x^{12} y^{16} z^{20}\)

Step by step solution

01

Apply the Negative Power Rule

The task provided involves simplifying \(\left(\frac{x^{3} y^{4} z^{5}}{x^{-3} y^{-4} z^{-5}}\right)^{-2}\). The first step is to apply the negative power rule, which means taking the reciprocal of the expression. Thus: \(\left(\frac{x^{-3} y^{-4} z^{-5}}{x^{3} y^{4} z^{5}}\right)^{2}\)
02

Subtract the Exponents

Next, divide each pair of variables with the same base, subtracting the exponents. Thus, the expression: \((x^{-3 - 3} y^{-4 - 4} z^{-5 - 5})^2\) simplifies to \((x^{-6} y^{-8} z^{-10})^2\)
03

Apply the Power of a Power Rule

The next step entails multiplying the exponents since we are raising a power to a power. This gives \(x^{-12} y^{-16} z^{-20}\)
04

Apply the Negative Power Rule

Finally, simplify the expression by taking the reciprocal of the variables with negative exponents, which gives us \(x^{12} y^{16} z^{20}\)

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

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