Simplify: ( square root of -x / 4 ) + ( square root of -x /16 ) - ( square root of -x/64 ), where x is a positive real number Ans given: 5/8 square root of x. i

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Solution

$\sqrt{- \frac{x}{4}} + \sqrt{- \frac{x}{16}} - \sqrt{- \frac{x}{64}} = \sqrt{\frac{x}{4} i^{2}} + \sqrt{\frac{x}{16} i^{2}} - \sqrt{\frac{x}{64} i^{2}} [as, i^{2} = - 1] = \frac{\sqrt{x}}{2} i + \frac{\sqrt{x}}{4} i - \frac{\sqrt{x}}{8} i = \frac{4 \sqrt{x} i + 2 \sqrt{x} i - \sqrt{x} i}{8} = \frac{5 \sqrt{x} i}{8}$

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