1 / (√7 – √2) — Start with the original expression. The denominator contains square roots, making it complex to work with.

(1 / (√7 – √2)) × ((√7 + √2) / (√7 + √2)) — Multiply both the numerator and the denominator by the conjugate of (√7 – √2), which is (√7 + √2), to rationalize the denominator.

(√7 – √2) × (√7 + √2) — Focus on the denominator first and expand it using the distributive property.

√7 × √7 + √7 × √2 – √2 × √7 – √2 × √2 — Break down the multiplication step-by-step.

7 – 2 — Simplify the terms to get 7 – 2 in the denominator.

(√7 + √2) / 5 — The new expression after rationalizing the denominator.

√7 / 5 + √2 / 5 — Divide each term in the numerator by the denominator to get the final simplified expression.

\( \frac{1}{\sqrt{3} – \sqrt{5}} \) — Start with the original expression. The denominator contains square roots, making it complex to work with.

\( \frac{1}{\sqrt{3} – \sqrt{5}} \times \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} \) — Multiply both the numerator and the denominator by the conjugate of \( \sqrt{3} – \sqrt{5} \), which is \( \sqrt{3} + \sqrt{5} \), to rationalize the denominator.

\( (\sqrt{3} – \sqrt{5}) \times (\sqrt{3} + \sqrt{5}) \) — Focus on the denominator first and expand it using the distributive property.

\( \sqrt{3} \times \sqrt{3} + \sqrt{3} \times \sqrt{5} – \sqrt{5} \times \sqrt{3} – \sqrt{5} \times \sqrt{5} \) — Break down the multiplication step-by-step.

\( 3 – 5 \) — Simplify the terms to get \( 3 – 5 \) in the denominator.

\( \frac{\sqrt{3} + \sqrt{5}}{3 – 5} \) — The new expression after rationalizing the denominator.

\( \frac{\sqrt{3} + \sqrt{5}}{-2} \) — Simplify the denominator \( 3 – 5 \) to \( -2 \).

\( \frac{\sqrt{3}}{-2} \) — Divide the first term in the numerator by the denominator.

\( \frac{\sqrt{5}}{-2} \) — Divide the second term in the numerator by the denominator.

\( -\frac{\sqrt{3}}{2} – \frac{\sqrt{5}}{2} \) — Combine the terms to get the final simplified expression.