/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 \(\frac{(a+b)^{-3}}{(a+b)^{-4}}\... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 73

\(\frac{(a+b)^{-3}}{(a+b)^{-4}}\)

Short Answer

Expert verified
(a + b)

Step by step solution

01

Understand the Problem

The given expression is \(\frac{(a+b)^{-3}}{(a+b)^{-4}}\). This involves division of two exponential expressions with the same base, \(a + b\).
02

Apply the Laws of Exponents

Use the property of exponents that states \(\frac{x^m}{x^n} = x^{m-n}\). Here, \(m = -3\) and \(n = -4\).
03

Simplify the Exponent

Calculate the new exponent by subtracting the exponents of the numerator and denominator: \(-3 - (-4)\). This simplifies to \(-3 + 4 = 1\).
04

Write the Simplified Expression

The simplified expression is \( (a+b)^1 \), which is just \(a + b\).

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Key Concepts

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

Exponential Expressions
Exponential expressions multiply a base by itself a certain number of times. For example, in the expression \(x^n\), \(x\) is the base and \(n\) is the exponent. This tells us to multiply \(x\) by itself \(n\) times.\
Exponents simplify expressions and show relationships between numbers. You often see them in algebra and other mathematical contexts. Developing a strong understanding of exponential expressions helps you work more efficiently with a wide range of math problems.\
When working with exponential expressions, remember these key points:\
    \
  • \(x^0 = 1\) for any non-zero \(x\).\
  • \(x^1 = x\).\
  • \(x^{-n} = \frac{1}{x^n}\) for any positive \(n\).\
Simplifying Exponents
Simplifying exponents makes complex expressions easier to work with. The rules of exponents help us do this. One of the most important rules is: \[ x^m \cdot x^n = x^{m+n} \], which combines the exponents when multiplying like bases.\
Likewise, when dividing with the same base, you subtract the exponents: \[ \frac{x^m}{x^n} = x^{m-n} \].\
For example, \frac{(a+b)^{-3}}{(a+b)^{-4}}\ was simplified by applying this rule: \frac{(a+b)^{-3}}{(a+b)^{-4}} = (a+b)^{-3 -(-4)} = (a+b)^{1} = a+b.\
This method brings a long or complex expression into a simpler form, making it easier to handle.
Division of Exponents
The division of exponents is seamless if you understand the laws of exponents. The key idea is to subtract the exponents of the same base. This is captured in the rule: \[ \frac{x^m}{x^n} = x^{m-n} \].\
Here, ensure that the bases of the exponents you're working with are the same, otherwise the rule doesn't apply.\
Taking our example \frac{(a+b)^{-3}}{(a+b)^{-4}}\, we noticed both the numerator and denominator had the same base, \(a + b\). Therefore, to simplify, we subtract: \-3 - (-4) = -3 + 4 = 1\
This simplifies the expression to: \( (a+b)^1 = a+b \)

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Most popular questions from this chapter

Perform each indicated operation. \(\left(16 x^{3}-x^{2}+3 x\right)+\left(-12 x^{3}+3 x^{2}+2 x\right)\)

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. $$ 103 \times 97 $$

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. $$ 201 \times 199 $$

Find each product. $$ (4 x+3)(2 y-1) $$

Find each product. $$ -3 r(r-1)\left(r^{2}+r+1\right) $$
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