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Chapter 7: Problem 24

Multiply as indicated. $$\frac{(x+4)^{3}}{(x+2)^{3}} \cdot \frac{x^{2}+4 x+4}{x^{2}+8 x+16}$$

Short Answer

Expert verified
The simplified form of the given expression is \((x+4) / ((x+2)*(x+2))\).

Step by step solution

01

Recognize the perfect squares and cubes

The expression in the numerator \((x+4)^3\) is a perfect cube, and the expression \((x+2)^3\) in the denominator is also a perfect cube. Similarly, \(x^{2}+4x+4\) is a perfect square of \((x+2)\) and \(x^{2}+8x+16\) is a perfect square of \((x+4)\).
02

Simplify the expressions

So, the given expression can be rewritten as: \((x+4)^3 / (x+2)^3 * (x+2)^2 / (x+4)^2\). Now, cancel \((x+4)^2\) from the numerator and denominator, result is \((x+4)/(x+2)\). Similarly, cancel \((x+2)^2\) from the numerator and denominator, and the result is \(1/(x+2)\).
03

Combine the results

So, the simplified form of the given expression is \((x+4) / (x+2) * 1 / (x+2)\). Multiply these two, result is \((x+4) / ((x+2)*(x+2))\).

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Most popular questions from this chapter

add or subtract as indicated. Simplify the result, if possible. $$\frac{3 y^{2}-2}{3 y^{2}+10 y-8}-\frac{y+10}{3 y^{2}+10 y-8}-\frac{y^{2}-6 y}{3 y^{2}+10 y-8}$$

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