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Chapter 0: Problem 22

Multiply or divide as indicated. $$\frac{x^{2}+6 x+9}{x^{3}+27} \cdot \frac{1}{x+3}$$

Short Answer

Expert verified
\The resultant expression after multiplication and division is: \(\frac{x+3}{x^2-3x+9}\)

Step by step solution

01

Factorize the Expressions

Begin by factorizing the given expressions. The numerator \(x^{2}+6x+9\) is a perfect square trinomial which can be factored as \((x+3)^2\), and the denominator \(x^{3}+27\) is a sum of cubes which can be factored as \((x+3)(x^2-3x+9)\). The expression now becomes: \(\frac{(x+3)^2}{(x+3)(x^2-3x+9)} \cdot \frac{1}{x+3}\)
02

Cancel Out Common Terms

After factorizing, cross-multiply and simplify by cancelling out the common terms. In this case, cancel out \(x+3\) from numerator and denominator. This results in the expression: \(\frac{x+3}{x^2-3x+9}\)
03

Simplify the Expression Further

The expression \(\frac{x+3}{x^2-3x+9}\) is the final simplified form of the given expression. It cannot be simplified further using factorization.

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Most popular questions from this chapter

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

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. $$ \frac{x^{2}+6 x+5}{x^{2}-25} $$

Simplify by reducing the index of the radical. $$\sqrt[9]{x^{6}}$$

Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. $$\text { a. } 3^{\frac{1}{2}} \square 3^{\frac{1}{3}}$$ $$\text { b. } \sqrt{7}+\sqrt{18} \square \sqrt{7}+18$$

Simplify using properties of exponents. $$\left(7 x^{\frac{1}{3}}\right)\left(2 x^{\frac{1}{4}}\right)$$
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