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Chapter 2: Problem 39

Simplify. \(\left(\frac{12 w^{5}}{4 x^{3} y^{6}}\right)^{2}\)

Short Answer

Expert verified
The simplified expression is \(\frac{9 w^{10}}{x^{6} y^{12}}\).

Step by step solution

01

Raise the entire fraction to the power of 2

We will start by applying the exponent to the entire fraction: \(\left(\frac{12 w^{5}}{4 x^{3} y^{6}}\right)^{2} = \frac{(12 w^{5})^{2}}{(4 x^{3} y^{6})^{2}}\)
02

Apply the Power of a Power Rule to each term

The Power of a Power Rule states that when raising powers to powers, we should multiply the exponents. Thus, apply this rule to every term in the numerator and denominator: \(\frac{(12 w^{5})^{2}}{(4 x^{3} y^{6})^{2}} = \frac{12^2 (w^{5})^2}{4^2 (x^{3})^2 (y^{6})^2}\)
03

Simplify the numerators and denominators

Now, carry out the calculations in the numerator and denominator: \(\frac{12^2 (w^{5})^2}{4^2 (x^{3})^2 (y^{6})^2} = \frac{144 w^{10}}{16 x^{6} y^{12}}\)
04

Simplify the fraction by cancelling out common factors

Both the numerator (144) and the denominator (16) have a common factor of 16. Divide the numerator and the denominator by 16 to simplify the fraction: \(\frac{144 w^{10}}{16 x^{6} y^{12}} = \frac{9 w^{10}}{x^{6} y^{12}}\) The simplified expression is: \(\frac{9 w^{10}}{x^{6} y^{12}}\)

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