Chapter O, Problem 1DE

To determine

To calculate: The expression $\sqrt{-5}$ in term of $i$.

Answer to Problem 1DE

Solution:

The expression $\sqrt{-5}$ in term of $i$ is $\underset{\_}{\sqrt{5}i}$.

Explanation of Solution

Given Information:

The expression is $\sqrt{-5}$.

Formula used:

Complex number:

The square root of a $-1$ is represented as $i$. The term $i$ is used to represent a negative number.

Mathematically,

$\sqrt{-1}=i$

To write any complex number in form of $i$, following steps are follows:

$\begin{array}{c}\sqrt{-p}=\sqrt{-1\cdot p}\\ =\sqrt{-1}\cdot \sqrt{p}\\ =i\sqrt{p}\\ =\sqrt{p}i\end{array}$

Here, $p$ is a positive real number.

Calculation:

Consider the expression.

$\sqrt{-5}$

Simplify as

$\begin{array}{c}\sqrt{-5}=\sqrt{-1\cdot 5}\\ =\sqrt{-1}\cdot \sqrt{5}\end{array}$

Substitute $i$ for $\sqrt{-1}$.

$\begin{array}{c}\sqrt{-5}=i\cdot \sqrt{5}\\ =\sqrt{5}i\end{array}$

Therefore, the complex number $\sqrt{-5}$ in term of $i$ is $\sqrt{5}i$.