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Chapter 0: Problem 114

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\frac{\left(2^{-1} x^{-3} y^{-1}\right)^{-2}\left(2 x^{-6} y^{4}\right)^{-2}\left(9 x^{3} y^{-3}\right)^{0}}{\left(2 x^{-4} y^{-6}\right)^{2}}$$

Short Answer

Expert verified
The simplified expression is \(16 y^6 / x^{18}\)

Step by step solution

01

Simplify Negative Exponents

The first step is to simplify the negative exponents in the expression. A negative exponent translates to taking the reciprocal of the base. Thus, the expression:\n\( \left(2^{-1} x^{-3} y^{-1}\right)^{-2} \left(2 x^{-6} y^{4}\right)^{-2} \left(9 x^{3} y^{-3}\right)^{0} / \left(2 x^{-4} y^{-6}\right)^{2} \) \nbecomes \n\( \left({1/2} x^{3} y\right)^{-2} \left({1/2} x^{6} y^{-4}\right)^{-2} \left(9 x^{3} y^{-3}\right)^{0} / \left({1/2} x^{4} y^{6}\right)^{2} \)
02

Simplify Expressions Raised to 0 power

Any number raised to the power of zero is 1. Thus, the expression above simplifies even further to become:\n \( \left({1/2} x^{3} y\right)^{-2} \left({1/2} x^{6} y^{-4}\right)^{-2} / \left({1/2} x^{4} y^{6}\right)^{2} \)
03

Simplify Expressions Raised to Negative Power

The negative power translates to reciprocal of expression raised to the power. Therefore, the expression simplifies to: \n\(1/\left({1/2} x^{3} y\right)^2 1/\left({1/2} x^{6} y^{-4}\right)^2 / \left({1/2} x^{4} y^{6}\right)^2\) = \n\({4x^{-6} y^{-2}}{4x^{-12} y^8}/{4x^8 y^{12}}\)
04

Simplify the expression

Now by simplifying the above expression we can cancel out the terms. We get: \n\( \frac{4x^{-6} y^{-2} * 4 x^{-12} y^8}{4x^8 y^{12}}\) = \n\(16 x^{-18} y^6\) = \n\(16 y^6 / x^{18}\)

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Most popular questions from this chapter

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \% .\) Now the sale price is reduced by another \(30 \%\). If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?

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