QuestionDecember 14, 2025

27) Rationalize the denominator to simplify the following expression. (4sqrt (15))/(1+2sqrt (5))

27) Rationalize the denominator to simplify the following expression. (4sqrt (15))/(1+2sqrt (5))
27) Rationalize the denominator to simplify the following expression.
(4sqrt (15))/(1+2sqrt (5))

Solution
4.5(242 votes)

Answer

\frac{40\sqrt{3} - 4\sqrt{15}}{19} Explanation 1. Multiply by the conjugate Multiply numerator and denominator by 1 - 2\sqrt{5}. 2. Expand the numerator 4\sqrt{15}(1 - 2\sqrt{5}) = 4\sqrt{15} - 8\sqrt{75} 3. Simplify \sqrt{75} \sqrt{75} = 5\sqrt{3}, so -8\sqrt{75} = -40\sqrt{3} 4. Write simplified numerator 4\sqrt{15} - 40\sqrt{3} 5. Expand the denominator using (a+b)(a-b) = a^2 - b^2 (1 + 2\sqrt{5})(1 - 2\sqrt{5}) = 1^2 - (2\sqrt{5})^2 = 1 - 4 \times 5 = 1 - 20 = -19 6. Combine results \frac{4\sqrt{15} - 40\sqrt{3}}{-19} 7. Simplify sign -\frac{4\sqrt{15} - 40\sqrt{3}}{19} = \frac{40\sqrt{3} - 4\sqrt{15}}{19}

Explanation

1. Multiply by the conjugate<br /> Multiply numerator and denominator by $1 - 2\sqrt{5}$.<br /><br />2. Expand the numerator<br /> $4\sqrt{15}(1 - 2\sqrt{5}) = 4\sqrt{15} - 8\sqrt{75}$<br /><br />3. Simplify $\sqrt{75}$<br /> $\sqrt{75} = 5\sqrt{3}$, so $-8\sqrt{75} = -40\sqrt{3}$<br /><br />4. Write simplified numerator<br /> $4\sqrt{15} - 40\sqrt{3}$<br /><br />5. Expand the denominator using $(a+b)(a-b) = a^2 - b^2$<br /> $(1 + 2\sqrt{5})(1 - 2\sqrt{5}) = 1^2 - (2\sqrt{5})^2 = 1 - 4 \times 5 = 1 - 20 = -19$<br /><br />6. Combine results<br /> $\frac{4\sqrt{15} - 40\sqrt{3}}{-19}$<br /><br />7. Simplify sign<br /> $-\frac{4\sqrt{15} - 40\sqrt{3}}{19} = \frac{40\sqrt{3} - 4\sqrt{15}}{19}$
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