Question

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

$\frac{2+\sqrt{-8}}{1+\sqrt{-2}}$

Short answer

The value of radical expression is $2+0i$.

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

$$\begin{gathered} \frac{2+\sqrt{-8}}{1+\sqrt{-2}}=\frac{2+\sqrt{8}i}{1+\sqrt{2}i} \\ =\frac{2+2\sqrt{2}i}{1+\sqrt{2}i}\cdot \frac{1-\sqrt{2}i}{1-\sqrt{2}i} \\ =\frac{\left(2+2\sqrt{2}i\right)\left(1-\sqrt{2}i\right)}{\left(1+\sqrt{2}i\right)\left(1-\sqrt{2}i\right)} \\ =\frac{2-2\sqrt{2}i+2\sqrt{2}i-4i^2}{{\left(1\right)}^2+{\left(\sqrt{2}\right)}^2} \\ =\frac{2+0i-4\left(-1\right)}{1+2} \\ =\frac{2+4}{3} \\ =2 \end{gathered}$$

Thus, the value of radical expression is $2+0i$.