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$Fully simplify the expression below and write your answer as a single fraction. (6x^(2)-294)/(x^(2)+16 x+63)*(x^(2)-81)/(x-7) Answer:$

Fully simplify the expression below and write your answer as a single fraction. $\newline$ $\frac{6 x^{2}-294}{x^{2}+16 x+63} \cdot \frac{x^{2}-81}{x-7}$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

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

\newline

\frac{6 x^{2}-294}{x^{2}+16 x+63} \cdot \frac{x^{2}-81}{x-7}

\newline

Answer:

Factor Numerator and Denominator: First, factor the numerator and the denominator of the first fraction. $\newline$ $6x^2 - 294$ can be factored by taking out the common factor of $6$ , which gives us $6(x^2 - 49)$ . $\newline$ $x^2 + 16x + 63$ can be factored into $(x + 7)(x + 9)$ because $7 \times 9 = 63$ and $7 + 9 = 16$ .
Factor Second Fraction: Now, factor the numerator and the denominator of the second fraction. $x^2 - 81$ is a difference of squares and can be factored into $(x + 9)(x - 9)$ . $x - 7$ cannot be factored further.
Cancel Common Factors: Next, we can simplify the expression by canceling out common factors in the numerator and the denominator. $\newline$ The $(x + 9)$ term in the numerator of the first fraction and the denominator of the second fraction can be canceled out. $\newline$ The $(x - 7)$ term in the denominator of the first fraction and the numerator of the second fraction can be canceled out.
Simplify Remaining Expression: After canceling, we are left with: $\newline$ $(6 \times (x^2 - 49)) / (x + 7) \times 1 / 1$ $\newline$ Simplify the remaining expression: $\newline$ $6 \times (x - 7)(x + 7) / (x + 7)$ $\newline$ Now, we can cancel out the $(x + 7)$ term in the numerator and the denominator.
Final Simplified Form: After canceling the $(x + 7)$ term, we are left with: $\newline$ $6 \times (x - 7)$ $\newline$ This is the fully simplified form of the expression.

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