Question

score: 0 of 5 points
simplify each expression in exact form and rewrite complex exp
a. $sqrt{-27}=$
b. $sqrt{-8}=$
incorrect!
we kindly recommend reviewing the expressing square roots or alternatively, you may engage in a discussion with mathgpt

Explanation:

Step1: Rewrite the negative - number under the square - root

Recall that $\sqrt{-a}=\sqrt{a}\times\sqrt{- 1}$, where $\sqrt{-1}=i$. For $\sqrt{-27}$, we can write $-27=-1\times27$. So $\sqrt{-27}=\sqrt{-1\times27}=\sqrt{-1}\times\sqrt{27}$.

Step2: Simplify the square - root of 27

We know that $27 = 9\times3$, so $\sqrt{27}=\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3}$. Since $\sqrt{-1}=i$, then $\sqrt{-27}=3\sqrt{3}i$.

Step3: Rewrite the negative - number under the square - root for $\sqrt{-8}$

Write $-8=-1\times8$. So $\sqrt{-8}=\sqrt{-1\times8}=\sqrt{-1}\times\sqrt{8}$.

Step4: Simplify the square - root of 8

We know that $8 = 4\times2$, so $\sqrt{8}=\sqrt{4\times2}=\sqrt{4}\times\sqrt{2}=2\sqrt{2}$. Since $\sqrt{-1}=i$, then $\sqrt{-8}=2\sqrt{2}i$.

The given answers are correct.

Answer:

a. $\sqrt{-27}=3\sqrt{3}i$
b. $\sqrt{-8}=2\sqrt{2}i$