/*! 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 75 \(\frac{(x+2 y)^{-3}}{(x+2 y)^{-... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 75

\(\frac{(x+2 y)^{-3}}{(x+2 y)^{-5}}\)

Short Answer

Expert verified
(x + 2y)^2

Step by step solution

01

Understand the Problem

We have a fraction with the same base \(x + 2y\) but with different exponents. The expression is \[ \frac{(x+2 y)^{-3}}{(x+2 y)^{-5}} \].
02

Apply the Quotient Rule for Exponents

When dividing two expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator. This gives \( (x+2 y)^{-3 - (-5)} \).
03

Simplify the Exponent

Simplify \(-3 - (-5)\) to get \(-3 + 5 = 2\). The expression becomes \((x + 2y)^2\).

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Key Concepts

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

fraction with same base
When dealing with fractions that have the same base in both the numerator and the denominator, your job becomes a lot simpler. In this problem, the base \( x + 2y \) remains the same in both parts of the fraction.

This is important because it allows us to directly apply exponent rules. Instead of worrying about different bases, we can focus solely on the exponents.

You'll find this approach handy when you're simplifying algebraic expressions, as it streamlines the process. Remember that having the same base means you only need to deal with the exponents, simplifying your work significantly.
simplify exponents
We're looking at simplifying exponents here, which is key when you encounter the quotient rule for exponents.

The quotient rule states that when dividing two expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator. So, in this problem: \[ \frac{(x + 2y)^{-3}}{(x + 2y)^{-5}} \]

By applying this rule, we take the exponent \( -3 - (-5) \). Notice how subtracting a negative is the same as adding a positive. Thus, \(-3 - (-5)\) simplifies to \(-3 + 5\).
This gives us: \[(x + 2y)^{2}\]

Understanding this principle will make it much easier to handle similar algebraic expressions in the future.
algebraic expressions
Algebraic expressions embody variables, coefficients, and exponents all working together. The given problem \[ \frac{(x + 2y)^{-3}}{(x + 2y)^{-5}} \] is a compact form of multiple terms and operations.

Simplifying such expressions requires an understanding of algebraic rules and properties, as we've seen with the quotient rule.

The final simplified form \((x + 2y)^{2}\) encapsulates all the work we did. Algebraic expressions like these may seem complex at first, but they follow logical steps:
  • Identify the base
  • Apply the quotient rule for exponents
  • Simplify the result

Mastering these steps will make tackling algebra much easier, breaking down even the most complex-looking problems into manageable parts.

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Most popular questions from this chapter

Without actually performing the operations, mentally determine the coefficient of the \(x^{2}\) -term in the simplified form of $$\left(-4 x^{2}+2 x-3\right)-\left(-2 x^{2}+x-1\right)+\left(-8 x^{2}+3 x-4\right)$$

Add or subtract as indicated. \(\left(3 k^{2} h^{3}+5 k h+6 k^{3} h^{2}\right)-\left(2 k^{2} h^{3}-9 k h+k^{3} h^{2}\right)\)

The distance to Earth from Pluto is \(4.58 \times 10^{9} \mathrm{~km}\). Pioneer 10 transmitted radio signals from Pluto to Earth at the speed of light, \(3.00 \times 10^{5} \mathrm{~km}\) per sec. About how long (in seconds) did it take for the signals to reach Earth?

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. $$ 301 \times 299 $$

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