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Chapter 5: Problem 35

Use properties of exponents to simplify each expression. Express answers in exponential form with positive exponents only. Assume that any variables in denominators are not equal to zero. \(\left(\frac{x^{5}}{x^{2}}\right)^{-4}\)

Short Answer

Expert verified
\(1/x^{12}\)

Step by step solution

01

Apply the Power of a Power Rule

According to the Power of a Power Rule, we multiply the exponents when an exponent is raised to another exponent. So, \(\left(\frac{x^{5}}{x^{2}}\right)^{-4}\) becomes \((x^{5*-4})/(x^{2*-4})\).
02

Simplify the expression

Multiply the exponents: \(x^{-20}/x^{-8}\)
03

Apply the Quotient of Powers Rule

The Quotient of Powers Rule allows us to subtract the denominator exponent from the numerator exponent. So, \(x^{-20}/x^{-8}\) becomes \(x^{-20-(-8)}=x^{-12}\).
04

Convert to positive exponent

Our expression is \(x^{-12}\). We can convert it to positive exponent by flipping the base value to the denominator, leaving us with \(1/x^{12}\). This is the final expression in exponential form with a positive exponent.

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