Evaluate the expression $1 + \frac { \sqrt { - 36 } \cdot \sqrt { - 49 } i } { \sqrt { 21 } }$ and write the result in the form $a + b i$ .
A) $\sqrt { 21 } - 42 i$
B) $21 - 42 i$
C) $1 - 2 i \sqrt { 21 }$
D) $- \sqrt { 21 } + 42$
E) $- \sqrt { 21 } - 42 i$

Question

Quizplus · Accepted Answer

We begin by simplifying the expression inside the square root as follows:

$$\sqrt{-36} \cdot \sqrt{-49} = 6i \cdot 7i = -42$$

So the expression becomes:

$$1 + \frac{-42i}{\sqrt{21}}$$

To rationalize the denominator, we multiply the top and bottom of the fraction by $\sqrt{21}$:

$$1 - \frac{42i\sqrt{21}}{21}$$

Simplifying, we get:

$$1 - 2i\sqrt{21}$$

Therefore, our answer is $\boxed{\textbf{(C)}}$.

Evaluate the Expression 1+−36⋅−49i211 + \frac { \sqrt { - 36 } \cdot \sqrt { - 49 } i } { \sqrt { 21 } }1+21​−36​⋅−49​i​

Evaluate the Expression $1 + \frac { \sqrt { - 36 } \cdot \sqrt { - 49 } i } { \sqrt { 21 } }$