Question

How do you simplify $\sqrt { - 144}$?

Answer 1

We are given a negative number in square root and we have to find its square root. The square root of a negative number can be found by first splitting into two components. For example $\sqrt { - {a^2}}$ this can be split up like $\sqrt { - 1} \times \sqrt {{a^2}}$ because we cannot find the square root of the negative number so we split up and find the square root of the number separately. After finding the square root and multiplying it by the value of $\sqrt { - 1}$i.e. $i$ which is the complex form of a number so the square root of $\sqrt { - {a^2}} = \sqrt { - 1} \times \sqrt {{a^2}}$ on putting value we will get $i \times a = ai$. Similarly we will solve for the given number.

Complete step-by-step answer:
Step1: We are given a number i.e. $\sqrt { - 144}$ and we have to find its square root. So first we will split it into the multiplication of two numbers i.e.
$\Rightarrow \sqrt { - 144} = \sqrt { - 1} \times \sqrt {144}$
Then we will find the square root of the number $144$ then we get:
$\Rightarrow \sqrt { - 144} = \sqrt { - 1} \times 12$
Step2: As we know that the value of$\sqrt { - 1}$ is $i$ on putting the value of $\sqrt { - 1}$ we get:
$\Rightarrow \sqrt { - 144} = i \times 12$
On multiplication we will get:
$12i$

Step3: Final answer: Hence the square root is $12i$

Note:
In this type of question students mainly make one mistake that they do not split up the number into the form of product of two number i.e. $\sqrt { - 1}$ and other numbers. They sometimes directly find the square without taking the $- 1$ out from the square root. But we cannot do this; the proper way to find the square root is to split up into the product of $\sqrt { - 1}$ and the number. The number in square root is the form of the complex number that’s why we convert the square root of a negative number into the complex number form.