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Chapter 5: Problem 130

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(3 x^{-4} y z^{-7}\right)(3 x)^{-3}$$

Short Answer

Expert verified
The simplified form of the given expression is \(81y / (x^7 z^7)\).

Step by step solution

01

Rewrite the expression with clear exponent rules

First, we rewrite the given expression \((3 x^{-4} y z^{-7})(3 x)^{-3}\) as \(3^1 x^{-4} y^1 z^{-7} \times 3^3 x^{-3}\).
02

Combine like terms

Next, we combine terms with the same base. We have two terms with base 3, two with base \(x\), and one each of base \(y\) and base \(z\).
03

Apply properties of exponents

Given two terms with the same base, we add the exponents when we multiply them. Thus the term for base 3 becomes \(3^{1+3} = 3^4\). Likewise the term for base \(x\) becomes \(x^{-4-3} = x^{-7}\). The expressions with bases \(y\) and \(z\) remain unchanged as \(y^1\) and \(z^{-7}\).
04

Write the final simplified expression

Finally, we can write out the simplified expression. We bring the terms with negative exponents to the denominator and positive exponents stay in the numerator: \(3^4y = 81y\) in the numerator and \(x^7z^7\) in the denominator. Thus the final simplified form is \(81y / (x^7 z^7)\).

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Most popular questions from this chapter

Discuss situations in which a vertical format, rather than a horizontal format, is useful for multiplying polynomials.

Exercises \(110-112\) will help you prepare for the material covered in the next section. In each exercise, perform the long division without using a calculator, and then state the quotient and the remainder. $$1 9 \longdiv { 4 9 4 }$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for \(2^{-1}+2^{-2}\) of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained.

Use a vertical format to find each product. $$\begin{array}{r}7 x^{3}-5 x^{2}+6 x \\\3 x^{2}-4 x \\\\\hline\end{array}$$
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