Question - Rationalizing the Denominator of a Complex Fraction
Solution:
To express the given expression in the form $$ q + r \sqrt{s} $$, where $$ q, r, $$ and $$ s $$ are rational numbers, we need to rationalize the denominator. The given expression is:\[ \frac{2\sqrt{5} + 5\sqrt{2}}{2\sqrt{5} - 5\sqrt{2}} \]First, we can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square roots in the denominator. The conjugate of $$2\sqrt{5} - 5\sqrt{2}$$ is $$2\sqrt{5} + 5\sqrt{2}$$. Multiplying both the numerator and the denominator by the conjugate, we get:\[ \frac{(2\sqrt{5} + 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})}{(2\sqrt{5} - 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})} \]Now expand both the numerator and the denominator:The numerator, when multiplied out, becomes:\[ (2\sqrt{5} \cdot 2\sqrt{5}) + (2\sqrt{5} \cdot 5\sqrt{2}) + (5\sqrt{2} \cdot 2\sqrt{5}) + (5\sqrt{2} \cdot 5\sqrt{2}) \]\[ = 4 \cdot 5 + 2 \cdot 5 \cdot \sqrt{10} + 5 \cdot 2 \cdot \sqrt{10} + 25 \cdot 2 \]\[ = 20 + 10\sqrt{10} + 10\sqrt{10} + 50 \]\[ = 70 + 20\sqrt{10} \]The denominator, when multiplied out, becomes a difference of squares:\[ (2\sqrt{5})^2 - (5\sqrt{2})^2 \]\[ = 4 \cdot 5 - 25 \cdot 2 \]\[ = 20 - 50 \]\[ = -30 \]Combining the expanded numerator and the expanded denominator, we have:\[ \frac{70 + 20\sqrt{10}}{-30} \]Now, simplify this by dividing both terms in the numerator by the denominator:\[ \frac{70}{-30} + \frac{20\sqrt{10}}{-30} \]\[ = -\frac{7}{3} - \frac{2\sqrt{10}}{3} \]So the final answer, expressed in the form $$ q + r \sqrt{s} $$, is:\[ q = -\frac{7}{3}, \quad r = -\frac{2}{3}, \quad s = 10 \]
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