Question

choose all the true statements.
a $sqrt{-2} = sqrt{2}$
b $sqrt{-2} = sqrt{2}i$
c $sqrt{-8} = 2isqrt{2}$
d $sqrt{-8} = -2sqrt{2}$
e $sqrt{-16} = 4i$
f $sqrt{-16} = -4$

Explanation:

Step1: Recall imaginary unit definition

By definition, $i = \sqrt{-1}$, so $\sqrt{-a} = i\sqrt{a}$ for $a>0$.

Step2: Evaluate Option A

$\sqrt{-2} = i\sqrt{2}
eq \sqrt{2}$, so A is false.

Step3: Evaluate Option B

$\sqrt{-2} = i\sqrt{2} = \sqrt{2}i$, so B is true.

Step4: Evaluate Option C

$\sqrt{-8} = i\sqrt{8} = i\sqrt{4\times2} = 2i\sqrt{2}$, so C is true.

Step5: Evaluate Option D

$\sqrt{-8} = 2i\sqrt{2}
eq -2\sqrt{2}$, so D is false.

Step6: Evaluate Option E

$\sqrt{-16} = i\sqrt{16} = 4i$, so E is true.

Step7: Evaluate Option F

$\sqrt{-16} = 4i
eq -4$, so F is false.

Answer:

B. $\sqrt{-2}=\sqrt{2}i$, C. $\sqrt{-8}=2i\sqrt{2}$, E. $\sqrt{-16}=4i$