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Chapter 5: Problem 131

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

Step by step solution

01

Distribute the exponent outside the parentheses to each term inside

Utilize the rule of exponents that says \((a*b)^n = a^n * b^n\). Here, the \(-2\) exponent applies to every term in the parentheses. The expression therefore becomes: \[x^{3*(-2)}y^{4*(-2)}z^{5*(-2)}\] \[x^{-6}y^{-8}z^{-10}\] Similarly for the denominator, it becomes: \[x^{(-3)*(-2)}y^{(-4)*(-2)}z^{(-5)*(-2)}\] \[x^{6}y^{8}z^{10}\]
02

Simplify the fractions by applying the rule of exponents for division

When you divide with the same base, subtract the exponent in the denominator from the exponent in the numerator. Therefore, the simplified expression becomes: \[\frac{x^{-6}}{x^{6}}\frac{y^{-8}}{y^{8}}\frac{z^{-10}}{z^{10}}\] which simplifies to: \[x^{-6-6}y^{-8-8}z^{-10-10}\] \[x^{-12}y^{-16}z^{-20}\]
03

Convert the negative exponents to positive

Negative exponents indicate the reciprocal. Therefore, \(x^{-12}\) is \(\frac{1}{x^{12}}\), and the same applies to the remaining variables. The final simplified expression is hence: \[\frac{1}{x^{12}y^{16}z^{20}}\].

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