Question

How do we write the given expression in terms of $i$ : $\sqrt { - 45} $ ?

Answer 1

Hint: To solve this question, first we will try to observe the given expression by removing the negative sign from the expression, and then we will discuss the term how $\sqrt { - 1} $ is related to $i$. And finally, simplify the positive square root expression to get the final answer.

Complete step by step solution:
The given special symbol, $i$, is used to represent the square root of negative 1, $\sqrt { - 1} $.
As we know, there is no such thing in the real number universe as the $\sqrt { - 1} $ because there are no two identical numbers that we can multiply together to get -1 as the exact solution.
$1.1 = 1$ and $ - 1. - 1$ is also 1. Obviously $1. - 1 = - 1$ , but 1 and -1 are not the same number. They both have the same magnitude(distance from zero), but they are not identical.
So, when we have a number that involves a negative square root, math developed a plan to get around that problem by saying that anytime we run across that issue, we make our number positive so we can deal with it and put an $i$ at the end.
So, in this case:
$
  \sqrt { - 45} \\
   = \sqrt {45} \times \sqrt { - 1} \\
    \\
 $
As $i = \sqrt { - 1} $ :
$\therefore \sqrt { - 45} = \sqrt {45} i$
Since, $45 = 9 \times 5$ , the answer can be simplified to:
$\because \sqrt {45} i = \sqrt {9.5} i = 3\sqrt 5 i$

Hence, the given expression in terms of $i$ is $3\sqrt 5 i$.

Note:
Iota, $i$, is a Greek letter that is widely used in mathematics to denote the imaginary part of a complex number. Let's say we have an equation: ${x^2} + 1 = 0$ . In this case, the value of $x$ will be the square root of -1, which is fundamentally not possible.