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Chapter 0: Problem 112

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(\frac{x^{4} y^{5} z^{6}}{x^{-4} y^{-5} z^{-6}}\right)^{-4}$$

Short Answer

Expert verified
\(\frac{1}{x^{32} y^{40} z^{48}}\)

Step by step solution

01

Apply the negative exponent rule

To start with, let's apply the negative exponent rule to the expression inside the parenthesis. In the denominator, all variables have negative powers. According to the negative exponent rule, we can rewrite them as reciprocals with positive powers. So, \(x^{-4} = 1/x^{4}\), \(y^{-5} = 1/y^{5}\), and \(z^{-6} = 1/z^{6}\). Thus, the expression can be rewritten as \(\frac{x^{4} y^{5} z^{6}}{(1/x^{4})(1/y^{5})(1/z^{6})}\).
02

Simplify the expression inside the parenthesis

Using the properties of division over multiplication we get \(\frac{x^{4}}{1/x^{4}} \cdot \frac{y^{5}}{1/y^{5}} \cdot \frac{z^{6}}{1/z^{6}}\). Now, we remember that when we are dividing two exponents with the same base we subtract the exponents. So, we get \(x^{4+4} \cdot y^{5+5} \cdot z^{6+6}\), which simplifies to: \(x^{8} \cdot y^{10} \cdot z^{12}\).
03

Apply the power of a power rule

Now, we take the power (-4) outside the parentheses for each of the variables. According to the power of a power rule, when we raise a power to a power we multiply the exponents: \(x^{8 \times -4} \cdot y^{10 \times -4} \cdot z^{12 \times -4}\), which simplifies to \(x^{-32} \cdot y^{-40} \cdot z^{-48}\).
04

Apply the negative exponents rule

All the variables have negative exponents so according to the negative exponent rule, we can rewrite them as reciprocals with positive powers. So, \(x^{-32} = 1/x^{32}\), \(y^{-40} = 1/y^{40}\), and \(z^{-48} = 1/z^{48}\), yielding the final simplified expression: \(\frac{1}{x^{32} y^{40} z^{48}}\).

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