Question

fill in each blank so that the resulting statement is true.\\(\sqrt{-99} = \underline{quadquad} \sqrt{99} = \underline{quadquad} \sqrt{9\cdot11} = \underline{quadquad} \sqrt{11}\\)\
fill in each answer box so that the resulting statement is true.\\(\sqrt{-99} = \square \sqrt{99} = \square \sqrt{9\cdot11} = \square \sqrt{11}\\)\\(\text{(simplify your answer.)}\\)

Explanation:

Step1: Simplify \(\sqrt{-99}\)

Recall the imaginary unit \(i = \sqrt{-1}\). So, \(\sqrt{-99}=\sqrt{-1\times99}=\sqrt{-1}\times\sqrt{99}=i\sqrt{99}\).

Step2: Simplify \(\sqrt{99}\)

Factor 99 as \(9\times11\), and use the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)). So, \(\sqrt{99}=\sqrt{9\times11}=\sqrt{9}\times\sqrt{11}\).

Step3: Simplify \(\sqrt{9}\)

Since \(\sqrt{9} = 3\) (because \(3^2 = 9\)).

Answer:

\(\sqrt{-99}=\boldsymbol{i}\sqrt{99}=\boldsymbol{\sqrt{9}}\sqrt{9\cdot11}=\boldsymbol{3}\sqrt{11}\)