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

(3x+3)/(9x^(2)+81 x+72)*(x^(2)-64)/(x^(2)-2x-48)
Answer:

Fully simplify the expression below and write your answer as a single fraction.\newline3x+39x2+81x+72x264x22x48 \frac{3 x+3}{9 x^{2}+81 x+72} \cdot \frac{x^{2}-64}{x^{2}-2 x-48} \newlineAnswer:

Full solution

Q. Fully simplify the expression below and write your answer as a single fraction.\newline3x+39x2+81x+72x264x22x48 \frac{3 x+3}{9 x^{2}+81 x+72} \cdot \frac{x^{2}-64}{x^{2}-2 x-48} \newlineAnswer:
  1. Factor Common Terms: First, factor each polynomial in the expression to see if any terms can be canceled out.\newlineStarting with the numerator of the first fraction, factor out the common factor of 33.\newline(3x+3)=3(x+1)(3x+3) = 3(x+1)
  2. Factor Numerator 11: Now, factor the denominator of the first fraction. The quadratic 9x2+81x+729x^2 + 81x + 72 can be factored into two binomials.\newline9x2+81x+72=(3x+9)(3x+8)9x^2 + 81x + 72 = (3x+9)(3x+8)
  3. Factor Denominator 11: Next, factor the numerator of the second fraction. The difference of squares x264x^2 - 64 can be factored into (x+8)(x8)(x+8)(x-8). \newlinex264=(x+8)(x8)x^2 - 64 = (x+8)(x-8)
  4. Factor Numerator 22: Now, factor the denominator of the second fraction. The quadratic x22x48x^2 - 2x - 48 can be factored into two binomials.\newlinex22x48=(x8)(x+6)x^2 - 2x - 48 = (x-8)(x+6)
  5. Factor Denominator 22: Now that we have factored all parts of the expression, we can write the entire expression with these factors and look for common terms to cancel out.\newline3(x+1)3(x+9)(x+8)×(x+8)(x8)(x8)(x+6)\frac{3(x+1)}{3(x+9)(x+8)} \times \frac{(x+8)(x-8)}{(x-8)(x+6)}
  6. Combine Factors: We can cancel out the common terms (x+8)(x+8) and (x8)(x-8) from the numerator and denominator.\newlineThe expression simplifies to:\newline3(x+1)3(x+9)(x+6)\frac{3(x+1)}{3(x+9)(x+6)}
  7. Cancel Common Terms: We can also cancel out the common factor of 33 from the numerator and denominator.\newlineThe expression further simplifies to:\newlinex+1(x+9)(x+6)\frac{x+1}{(x+9)(x+6)}
  8. Further Simplify: Now we have the fully simplified expression as a single fraction.\newlineThe final answer is (x+1)/((x+9)(x+6))(x+1)/((x+9)(x+6)).

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