Questions: Rationalize the denominator and simplify. (5/(5+2√3))/((25/13)-(10√3/13))

Rationalize the denominator and simplify. (5/(5+2√3))/((25/13)-(10√3/13))
Transcript text: Rationalize the denominator and simplify. \[ \frac{\frac{5}{5+2 \sqrt{3}}}{\frac{25}{13}-\frac{10 \sqrt{3}}{13}} \]
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Solution

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Solution Steps

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial \(a + b\) is \(a - b\). This will eliminate the square root in the denominator. After rationalizing, simplify the expression by performing the arithmetic operations.

Step 1: Define the Expression

We start with the expression

\[ \frac{\frac{5}{5 + 2\sqrt{3}}}{\frac{25}{13} - \frac{10\sqrt{3}}{13}}. \]

Step 2: Rationalize the Denominator

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is

\[ \frac{25}{13} + \frac{10\sqrt{3}}{13}. \]

This gives us:

\[ \frac{5 \left( \frac{25}{13} + \frac{10\sqrt{3}}{13} \right)}{\left( \frac{25}{13} - \frac{10\sqrt{3}}{13} \right) \left( \frac{25}{13} + \frac{10\sqrt{3}}{13} \right)}. \]

Step 3: Simplify the Expression

After performing the multiplication and simplification, we find that the entire expression simplifies to

\[ 1. \]

Final Answer

Thus, the final result is

\[ \boxed{1}. \]

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