QuestionJuly 15, 2025

Rationalize the denominator and simplify. (sqrt (5)+sqrt (3))/(sqrt (5)-sqrt (3)) square

Rationalize the denominator and simplify. (sqrt (5)+sqrt (3))/(sqrt (5)-sqrt (3)) square
Rationalize the denominator and simplify. (sqrt (5)+sqrt (3))/(sqrt (5)-sqrt (3)) square

Solution
4.1(289 votes)

Answer

4 + \sqrt{15} Explanation 1. Multiply by the Conjugate Multiply numerator and denominator by the conjugate of the denominator, \sqrt{5}+\sqrt{3}. \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} \times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}} = \frac{(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5})^2-(\sqrt{3})^2} 2. Simplify the Denominator Use **difference of squares**: (a-b)(a+b) = a^2 - b^2. (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 3. Expand the Numerator Expand (\sqrt{5}+\sqrt{3})^2 using **(a+b)^2 = a^2 + 2ab + b^2**. (\sqrt{5})^2 + 2\sqrt{5}\sqrt{3} + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} 4. Combine Results Divide the expanded numerator by the simplified denominator. \frac{8 + 2\sqrt{15}}{2} = \frac{8}{2} + \frac{2\sqrt{15}}{2} = 4 + \sqrt{15}

Explanation

1. Multiply by the Conjugate<br /> Multiply numerator and denominator by the conjugate of the denominator, $\sqrt{5}+\sqrt{3}$.<br />$$ \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} \times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}} = \frac{(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5})^2-(\sqrt{3})^2} $$<br /><br />2. Simplify the Denominator<br /> Use **difference of squares**: $(a-b)(a+b) = a^2 - b^2$.<br />$$ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 $$<br /><br />3. Expand the Numerator<br /> Expand $(\sqrt{5}+\sqrt{3})^2$ using **$(a+b)^2 = a^2 + 2ab + b^2$**.<br />$$ (\sqrt{5})^2 + 2\sqrt{5}\sqrt{3} + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} $$<br /><br />4. Combine Results<br /> Divide the expanded numerator by the simplified denominator.<br />$$ \frac{8 + 2\sqrt{15}}{2} = \frac{8}{2} + \frac{2\sqrt{15}}{2} = 4 + \sqrt{15} $$
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