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Chapter 0: Problem 60

Simplify each exponential expression. $$\left(\frac{3 x^{4}}{y}\right)^{-3}$$

Short Answer

Expert verified
\(\frac{y^3}{27x^{12}}\)

Step by step solution

01

Apply the Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^{n}}\). Applying this to our expression gives: \(\left(\frac{3 x^{4}}{y}\right)^{-3} = \frac{1}{\left(\frac{3 x^{4}}{y}\right)^{3}}\)
02

Apply the Power of a Quotient Rule

This rule states that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). In this step apply it: \(\frac{1}{\left(\frac{3 x^{4}}{y}\right)^{3}} = \frac{1}{\left(3^{3} x^{4*3} y^{-1*3}\right)}\) which simplifies to \(\frac{1}{27x^{12}y^{-3}}\)
03

Apply the Negative Exponent Rule Again

The negative exponent rule is applied again to manage \(y^{-3}\): \(\frac{1}{27x^{12}y^{-3}} = \frac{1}{27x^{12}} * \frac{1}{y^{-3}} = \frac{y^3}{27x^{12}}\)
04

Final Simplified Expression

No more simplification is possible, hence the final answer is: \(\frac{y^3}{27x^{12}}\)

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