"Evaluate the radical expression, and express the result..." solved and explained

Lipossig

Answered question

2021-09-17

Evaluate the radical expression, and express the result in the form $a + b i .$
$\frac{\sqrt{- 36}}{\sqrt{- 2} \sqrt{- 9}}$

Clelioo

Skilled2021-09-18Added 88 answers

We have to evaluate the radical expression:

\frac{\sqrt{- 36}}{\sqrt{- 2} \sqrt{- 9}}

We know that

i = \sqrt{- 1} \Rightarrow i^{2} = - 1,

where i is the imaginary unit.
therefore,

\sqrt{- 36} = i \sqrt{36}

= 6 i

\sqrt{- 2} = i \sqrt{2}

\sqrt{- 9} = i \sqrt{9}

= 3 i

Applying above result for the given expression,

\frac{\sqrt{- 36}}{\sqrt{- 2} \sqrt{- 9}} = \frac{6 i}{i \sqrt{2} \times 3 i}

= \frac{6 i}{3 \sqrt{2} i^{2}}

= \frac{2 i}{\sqrt{2} (- 1)}

= \frac{2 i}{- \sqrt{2}}

= - \frac{2 i}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}

= - \frac{2 i \times \sqrt{2}}{2}

= - \sqrt{2} i

Hence, required answer is

- \sqrt{2} i

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