Questions: Which of the following is the quotient of the rational expressions shown here? x/(x-6) ÷ (x+6)/x A. (x+6)/(x-6) B. x^2/(x^2-36) C. -1

Which of the following is the quotient of the rational expressions shown here? x/(x-6) ÷ (x+6)/x A. (x+6)/(x-6) B. x^2/(x^2-36) C. -1
Transcript text: Which of the following is the quotient of the rational expressions shown here? \[ \frac{x}{x-6} \div \frac{x+6}{x} \] A. $\frac{x+6}{x-6}$ B. $\frac{x^{2}}{x^{2}-36}$ C. -1
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Solution

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Solution Steps

Step 1: Rewrite the Division as Multiplication

To divide two rational expressions, rewrite the division as multiplication by the reciprocal of the second expression: \[ \frac{x}{x-6} \div \frac{x+6}{x} = \frac{x}{x-6} \times \frac{x}{x+6} \]

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \[ \frac{x \cdot x}{(x-6)(x+6)} = \frac{x^2}{x^2 - 36} \]

Step 3: Compare with the Given Options

Compare the result with the provided options:

  • A. \(\frac{x+6}{x-6}\)
  • B. \(\frac{x^{2}}{x^{2}-36}\)
  • C. -1

The correct answer matches option B.

Final Answer

\(\boxed{\frac{x^{2}}{x^{2}-36}}\)

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