If \frac{\sqrt{x+20}+\sqrt{x-1}}{\sqrt{x+20}-\sqrt{x-1}}=\frac{7}{3}, then what is the value of \sqrt{(x+20)(x-1)}?

Explanation

Applying componendo and dividendo, \frac{\sqrt{x+20}}{\sqrt{x-1}} = \frac{7+3}{7-3} = \frac{10}{4} = \frac{5}{2}. Squaring both sides: \frac{x+20}{x-1} = \frac{25}{4} \implies 4x + 80 = 25x - 25 \implies 21x = 105 \implies x = 5. Then \sqrt{(5+20)(5-1)} = \sqrt{25 \times 4} = 10.

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