Problem 24 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \((x+4)\).

Step 2: 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 \(1/(x+2)\).

Step 3: Combine the results

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