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Chapter 4: Problem 88

\(\frac{\left(m^{-8} n^{-4}\right)^{2}}{m^{2} n^{5}}\)

Short Answer

Expert verified
\frac{1}{m^{18} n^{13}}.

Step by step solution

01

Simplify the Exponent

First, simplify \(\frac{\big(m^{-8} n^{-4}\big)^{2}}{m^{2} n^{5}}\). Use the power of a power property \( (a^m)^n = a^{mn} \) to simplify the numerator. \((m^{-8})^2 = m^{-16}\) and \((n^{-4})^2 = n^{-8}\), so \(\big(m^{-8} n^{-4}\big)^2 = m^{-16} n^{-8}\).
02

Substitute and Simplify

Replace the original expression with the simplified form: \(\frac{m^{-16} n^{-8}}{m^{2} n^{5}}\).
03

Apply the Quotient of Powers Property

Use the quotient of powers property \(a^m / a^n = a^{m-n}\) to simplify the expression: \(m^{-16} / m^2 = m^{-16-2} = m^{-18}\) and \(n^{-8} / n^5 = n^{-8-5} = n^{-13}\).
04

Combine Results

Combine the results from the previous step to get the final simplified form: \(m^{-18} n^{-13}\).
05

Write Answer with Positive Exponents

To express the answer using positive exponents, use the property \(a^{-m} = \frac{1}{a^m}\): \(m^{-18} n^{-13} = \frac{1}{m^{18} n^{13}}\).

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Key Concepts

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

Power of a Power Property
In this step-by-step exercise, we encounter the power of a power property. This property states that for any base

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$\begin{aligned} &y=x^{2}-4\\\&\begin{array}{c|c}\hline x & y \\\\\hline-2 & \\\\\hline-1 & \\\\\hline 0 & \\\\\hline 1 & \\\\\hline 2 &\end{array}\end{aligned}$$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ -3 a(3 a+1)(a-4) $$

Graph each equation by completing the table of values. $$\begin{aligned} &y=x^{2}-9\\\&\begin{array}{c|c}\hline x & y \\\\\hline-2 & \\\\\hline-1 & \\\\\hline 0 & \\\\\hline 1 & \\\\\hline 2 &\end{array}\end{aligned}$$ $$

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. $$ 101 \times 99 $$

Perform each indicated operation. \(\left[\left(8 m^{2}+4 m-7\right)-\left(2 m^{2}-5 m+2\right)\right]-\left(m^{2}+m+1\right)\)
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