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$Fully simplify the expression below and write your answer as a single fraction. (2x-2)/(4x+4)*(x^(3)-64 x)/(x^(3)-9x^(2)+8x) Answer:$

Fully simplify the expression below and write your answer as a single fraction. $\newline$ $\frac{2 x-2}{4 x+4} \cdot \frac{x^{3}-64 x}{x^{3}-9 x^{2}+8 x}$ $\newline$ Answer:

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Fully simplify the expression below and write your answer as a single fraction.

\newline

\frac{2 x-2}{4 x+4} \cdot \frac{x^{3}-64 x}{x^{3}-9 x^{2}+8 x}

\newline

Answer:

Identify Common Factors: First, we should look for common factors in the numerators and denominators that can be simplified. Let's start by factoring each part of the expression.
Factor Out Common Factor: Factor out the common factor of $2$ in the first numerator and denominator: $\frac{2x-2}{4x+4} = \frac{2(x-1)}{4(x+1)}$ Now, we can simplify the factor of $2$ in the numerator with the factor of $4$ in the denominator: $\frac{2(x-1)}{4(x+1)} = \frac{x-1}{2(x+1)}$
Factor Difference of Cubes: Next, factor the second numerator by recognizing it as a difference of cubes: $x^3 - 64x = x(x^2 - 64) = x(x^2 - 8^2) = x(x + 8)(x - 8)$
Factor Quadratic Expression: Now, factor the second denominator by recognizing it as a quadratic expression that can be factored by grouping: $x^3 - 9x^2 + 8x = x(x^2 - 9x + 8) = x(x - 1)(x - 8)$
Combine Simplified Parts: Combine the simplified parts of the expression: $\left(\frac{x-1}{2(x+1)}\right) \cdot \left(\frac{x(x+8)(x-8)}{x(x-1)(x-8)}\right)$
Cancel Common Factors: Now, cancel out the common factors in the numerator and denominator: $\newline$ The $(x-1)$ terms cancel, one $x$ term cancels, and the $(x-8)$ terms cancel: $\newline$ $\frac{(x-1)}{2(x+1)} \times \frac{x(x+8)(x-8)}{x(x-1)(x-8)} = \frac{1}{2(x+1)} \times \frac{(x+8)}{1}$
Simplify Remaining Expression: Simplify the remaining expression: $(\frac{1}{2}(x+1)) \cdot (x+8) = \frac{x+8}{2(x+1)}$
Write Final Answer: The expression is now fully simplified, and we can write it as a single fraction: $\frac{x+8}{2(x+1)}$