Miguel Davenport

2022-01-30

How do you simplify $5\sqrt{-25}×i?$

euromillionsna

Expert

Step 1
$5\sqrt{-25}×i=5i\sqrt{25}×i=5i×5i=25×{i}^{2}=-25$
In common with all non-zero numbers, -25 has two square roots.
The square root represented by the symbols $\sqrt{-25}$ is the principal square root $i\sqrt{25}=5i$. The other square root is $-\sqrt{-25}=-i\sqrt{25}=-5i$
When then $\sqrt{ab}=\sqrt{a}\sqrt{b}$, but that fails if both $a<0$ and $b<0$ as in this example:
$1=\sqrt{1}=\sqrt{-1×-1}\ne \sqrt{-1}×\sqrt{-1}=-1$

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