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Chapter 0: Problem 29

Multiply or divide as indicated. $$\frac{x^{2}-25}{2 x-2} \div \frac{x^{2}+10 x+25}{x^{2}+4 x-5}$$

Short Answer

Expert verified
\(\frac{x-5}{2(x+5)}\)

Step by step solution

01

- Simplify each fraction

Firstly, try simplifying the given fractions. We can simplify \(\frac{x^{2}-25}{2x-2}\) by factoring and simplifying: \(\frac{(x-5)(x+5)}{2(x-1)}\). Similarly, we can simplify \(\frac{x^{2}+10x+25}{x^{2}+4x-5}\) to: \(\frac{(x+5)^2}{(x-1)(x+5)}\).
02

- Change the operation to multiplication

To divide fractions, it's convenient to change the operation to multiplication by using the reciprocal of the second fraction. So we rewrite the problem as: \(\frac{(x-5)(x+5)}{2(x-1)} \times \frac{(x-1)(x+5)}{(x+5)^2}\).
03

- Cancel out common factors

In the expression \(\frac{(x-5)(x+5)}{2(x-1)} \times \frac{(x-1)(x+5)}{(x+5)^2}\), the factors \((x-1)\), \((x+5)\) in the numerator and denominator can be cancelled to simplify the expression. Doing so, we get: \(\frac{x-5}{2} \times \frac{1}{x+5}\).
04

- Simplify the multiplication of fractions

Finally, multiply the fractions as \(\frac{x-5}{2} \times \frac{1}{x+5}\) which simplifies to \(\frac{x-5}{2(x+5)}\).

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Most popular questions from this chapter

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.

Simplify using properties of exponents. $$\left(x^{\frac{2}{3}}\right)^{3}$$

Simplify using properties of exponents. $$\left(25 x^{4} y^{6}\right)^{\frac{1}{2}}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Special-product formulas have patterns that make their multiplications quicker than using the FOIL method.

Perform the indicated operations. $$ [(7 x+5)+4 y][(7 x+5)-4 y] $$
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