Question

Find the square root of $ - 121$. $\left( {\sqrt { - 1} = i} \right)$
(A) $ - 11$
(B) $11$
(C) $11i$
(D) $ - 11i$

Answer 1

Hint:In this question we use some basic property of square and square root. In this we use $\sqrt {a \times b} = \sqrt a \times \sqrt b $ and $\sqrt {a \times a} = a$ and square root of any number can be found by prime factorization method.

Complete step-by-step answer:
Given is square root of $ - 121$
So this can be written as multiple of two number
So $ - 121 = - 1 \times 121$
Now taking square root both side we get
$\sqrt { - 121} = \sqrt { - 1 \times 121} $
By using property $\sqrt { - 1 \times 121} $ can be written as $\sqrt { - 1} \times \sqrt {121} $
$\because $ $\sqrt {a \times b} = \sqrt a \times \sqrt b $
From this we can write that
$\sqrt { - 121} = \sqrt { - 1} \times \sqrt {121} $
Now by using prime factorization method $121 = 11 \times 11$
So
$\sqrt { - 121} = \sqrt { - 1} \times \sqrt {11 \times 11} $
Now we know that $\sqrt {a \times a} = a$
By using this we can say
$\sqrt { - 121} = \sqrt { - 1} \times 11$
Now we know that $\sqrt { - 1} = i$ as mentioned in the question.
So $\sqrt { - 121} = i \times 11$
And finally we get our required answer that is
$\sqrt { - 121} = 11i$

So, the correct answer is “Option C”.

Note:The number $i$ is called an imaginary number. Generally, the square root of any negative real number is called an imaginary number. Like $\sqrt { - 1} = i$
If we square both sides we get ${i^2} = 1$ and a cube of this give ${i^3} = - i$. This function is generally used in complex number system and is called IOTA.
Definition of square and square root :
Finding the square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers. The square root of a number, n, written below is the number that gives n when multiplied by itself.
Some Properties of square root
1. The Square root of an even perfect square is even and that of an odd Perfect square is odd.
2. Since there is no number whose square is negative the square root of a negative number is not defined.
3. If a number ends with an odd number of zeroes, then it cannot have a square root which is a natural number.
4. If the units digit of a number is 2, 3, 7 or 8 then square root of that number (in natural numbers) is not possible.
5. If m is not a perfect square, then there is no integer n such that the square root of m is n.