How do you simplify $\sqrt{- \frac{4}{5}}$ $sqrt(-4/5)$ ?

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How do you simplify $\sqrt{- \frac{4}{5}}$ $sqrt(-4/5)$ ?

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Answer 1 · 2015-12-27T14:18:31

$\sqrt{- \frac{4}{5}} = i \sqrt{\frac{4}{5}} = \frac{2 \sqrt{5}}{5} i$ $sqrt(-4/5) = i sqrt(4/5) = (2sqrt(5))/5 i$

Explanation:

By definition and convention, if $x < 0$ $x < 0$ then $\sqrt{x} = i \sqrt{- x}$ $sqrt(x) = i sqrt(-x)$ where $i$ $i$ is the imaginary unit.

Also, if $a, b > 0$ $a, b > 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$ $sqrt(ab) = sqrt(a)sqrt(b)$

So:

$\sqrt{- \frac{4}{5}} = i \sqrt{\frac{4}{5}} = i \sqrt{\frac{4}{25} \cdot 5} = i \sqrt{\frac{4}{25}} \sqrt{5} = i \sqrt{{(\frac{2}{5})}^{2}} \sqrt{5}$ $sqrt(-4/5) = i sqrt(4/5) = i sqrt(4/25 * 5) = i sqrt(4/25)sqrt(5) = i sqrt((2/5)^2)sqrt(5)$

$= \frac{2 \sqrt{5}}{5} i$ $= (2sqrt(5))/5 i$

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