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Chapter 4: Problem 93

\(\frac{(2 r)^{-4}(2 r)^{5}}{(2 r)^{9}(2 r)^{-7}}\)

Short Answer

Expert verified
\( \frac{1}{2r} \)

Step by step solution

01

Combine Exponents in the Numerator

In the numerator, combine the exponents of the terms \( (2r)^{-4} \) and \( (2r)^5 \) using the property \( a^m \times a^n = a^{m+n} \). \[ (2r)^{-4} \times (2r)^5 = (2r)^{-4+5} = (2r)^1 = 2r \]
02

Combine Exponents in the Denominator

In the denominator, combine the exponents of the terms \( (2r)^9 \) and \( (2r)^{-7} \) using the same property. \[ (2r)^9 \times (2r)^{-7} = (2r)^{9-7} = (2r)^2 \]
03

Simplify the Fraction

Now, divide the combined terms from the numerator and the denominator: \[ \frac{2r}{(2r)^2} = (2r)^{1-2} = (2r)^{-1} \]
04

Simplify the Exponent

Simplify the negative exponent using the property \( a^{-n} = \frac{1}{a^n} \): \[ (2r)^{-1} = \frac{1}{2r} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Rules
Exponent rules help us simplify expressions involving powers. When multiplying like bases, add the exponents. For instance, if we have \((2r)^{-4} \times (2r)^5\), we combine the exponents: \

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Most popular questions from this chapter

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 76\. \(\frac{0.000000081(0.000036)}{0.00000048}\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 5 k^{2}\left(k^{3}-3\right)\left(k^{2}-k+4\right) $$

In \(2016,\) the U.S. government collected about \(\$ 5712\) per person in individual income taxes. If the population of the United States at that time was 323,000,000 , how much did the government collect in taxes for \(2016 ?\) (Data from www.usgovernmentrevenue.com)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (8 m+3 n)^{2} $$
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