Question

rationalize the denominator and simplify. \\(\frac{3}{5 - 2\sqrt{2}}\\)

Explanation:

Step1: Identify conjugate of denominator

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

Step2: Multiply numerators and denominators

Numerator: \(3\times(5 + 2\sqrt{2}) = 15 + 6\sqrt{2}\)
Denominator: Use \((a - b)(a + b)=a^2 - b^2\), so \((5)^2 - (2\sqrt{2})^2 = 25 - 8 = 17\)

Step3: Write the simplified form

\(\frac{15 + 6\sqrt{2}}{17}\)

Answer:

\(\frac{15 + 6\sqrt{2}}{17}\)