Question

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

$\left(3-\sqrt{-5}\right)\left(1+\sqrt{-1}\right)$

Short answer

The value of radical expression is $\left(3+\sqrt{5}\right)+\left(3-\sqrt{5}\right)i$.

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,

• localid="1644312669395" $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$
  
• localid="1644313235635" $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
  

Step 3. Evaluate the radical expression.

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

$$\begin{gathered} \left(3-\sqrt{-5}\right)\left(1+\sqrt{-1}\right)=\left(3-\sqrt{5}i\right)\left(1+i\right) \\ =3+3i-\sqrt{5}i-\sqrt{5}{i}^2 \\ =3+\left(3-\sqrt{5}\right)i-\sqrt{5}\left(-1\right) \\ =\left(3+\sqrt{5}\right)+\left(3-\sqrt{5}\right)i \end{gathered}$$

Thus, the value of radical expression is $\left(3+\sqrt{5}\right)+\left(3-\sqrt{5}\right)i$.