Question

Radical Expressions Evaluate the radical expression, and express the result in the form$a+bi$.

$\left(\sqrt{3}-\sqrt{-4}\right)\left(\sqrt{6}-\sqrt{-8}\right)$

Short answer

The value of radical expression is $-\sqrt{2}-4\sqrt{6}i$.

Step-by-step solution

Step 1. Concept of complex number.

A complex number is of the form $a+bi$. Where a and b are some real numbers and i is the imaginary number.

The number a is known as the real part of the complex number and b is known as imaginary part of the complex number.

Step 2. Concept of imaginary number.

The imaginary number i is defined as the square root of negative one. That is, $i=\sqrt{-1}$or $i^2=-1$.

The properties of radicals,

• $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$
  
• $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
  

Step 3. Evaluate the radical expression.

To evaluate the given radical expression, use the concept of imaginary number and simplify further as shown below

$$\begin{gathered} \left(\sqrt{3}-\sqrt{-4}\right)\left(\sqrt{6}-\sqrt{-8}\right)=\left(\sqrt{3}-\sqrt{4}i\right)\left(\sqrt{6}-\sqrt{8}i\right) \\ =\left(\sqrt{3}-2i\right)\left(\sqrt{6}-2\sqrt{2}i\right) \\ =\sqrt{18}-2\sqrt{6}i-2\sqrt{6}i+4\sqrt{2}{i}^2 \\ =3\sqrt{2}-4\sqrt{6}i+4\sqrt{2}\left(-1\right) \\ =3\sqrt{2}-4\sqrt{6}i-4\sqrt{2} \\ =-\sqrt{2}-4\sqrt{6}i \end{gathered}$$

Thus, the value of radical expression is $-\sqrt{2}-4\sqrt{6}i$.