Question

simplify (rationalize the denominator). assume that all variables represent positive real numbers.\\(
\frac{7}{3 - 2\sqrt{5}}\\)
\\(
\circ\\ \frac{21 + 14\sqrt{5}}{11}\\)
\\(
\circ\\ \frac{-21 - 14\sqrt{5}}{11}\\)
\\(
\circ\\ \frac{21 - 14\sqrt{5}}{11}\\)
\\(
\circ\\ \frac{21 + 14\sqrt{5}}{29}\\)
\\(
\circ\\ \frac{-21 - 14\sqrt{5}}{29}\\)

Explanation:

Step1: Identify conjugate

The denominator is \(3 - 2\sqrt{5}\), its conjugate is \(3 + 2\sqrt{5}\). Multiply numerator and denominator by the conjugate.

$$ \frac{7}{3 - 2\sqrt{5}} \times \frac{3 + 2\sqrt{5}}{3 + 2\sqrt{5}} $$

Step2: Multiply numerator

Multiply numerators: \(7\times(3 + 2\sqrt{5}) = 21 + 14\sqrt{5}\)

Step3: Multiply denominator

Multiply denominators using \((a - b)(a + b)=a^2 - b^2\): \((3)^2 - (2\sqrt{5})^2 = 9 - 20 = -11\)

Step4: Simplify fraction

Now we have \(\frac{21 + 14\sqrt{5}}{-11}\), factor out -1 from numerator and denominator: \(\frac{-(21 + 14\sqrt{5})}{11}=\frac{-21 - 14\sqrt{5}}{11}\)

Answer:

B. \(\frac{-21 - 14\sqrt{5}}{11}\) (assuming the second option is labeled B, with the text \(\frac{-21 - 14\sqrt{5}}{11}\))