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Chapter 0: Problem 17

Multiply or divide as indicated. $$\frac{x^{2}-9}{x^{2}} \cdot \frac{x^{2}-3 x}{x^{2}+x-12}$$

Short Answer

Expert verified
The simplified form of the expression is \(\frac{x+3}{x \cdot (x+4)}\).

Step by step solution

01

Simplify the Expressions

The numerator of first fraction \(x^{2}-9\) is a difference of squares and can be factored to \((x-3)(x+3)\). The denominator in the second fraction \(x^{2}+x-12\) has the roots \(x=-4, x=3\), which means it can be factored into \((x+4)(x-3)\). Substituting these into the original expression, we obtain \(\frac{(x-3)(x+3)}{x^{2}} \cdot \frac{x(x-3)}{(x+4)(x-3)}\).
02

Perform the Multiplication

Now multiply straight across. We multiply the numerators \( (x-3)(x+3) \) and \( x(x-3)\), and multiply the denominators \(x^{2}\) and \( (x+4)(x-3)\) to get \(\frac{(x-3)^2 \cdot x \cdot (x+3)}{x^{2} \cdot (x+4) \cdot (x-3)}\).
03

Simplify the Resulting Fraction

Now cancel out common factors from the numerator and the denominator. \((x-3)\) is common in the numerator and denominator and can be cancelled leaving \(\frac{x \cdot (x+3)}{x^{2} \cdot (x+4)}\).
04

Final Simplification

Finally, simplify by cancelling out the common factor \(x\) in the numerator and denominator, resulting in the simplest form of the expression: \(\frac{x+3}{x \cdot (x+4)}\).

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Most popular questions from this chapter

Evaluate each expression without using a calculator. $$8^{\frac{2}{3}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ x^{3}-64-(x+4)\left(x^{2}+4 x-16\right) $$

Will help you prepare for the material covered in the next section.Exercises \(144-146\) will help you prepare for the material covered in the next section. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$ \frac{x^{2}+6 x+5}{x^{2}-25} $$

Factor completely. $$ (x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}} $$

Simplify using properties of exponents. $$\frac{72 x^{\frac{3}{4}}}{9 x^{\frac{1}{3}}}$$
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