Question

What is the square root of \[ - 72\]?

Answer 1

Hint: Here, in the question given, we are asked to find the square root of \[ - 72\]. Now, we know that the square of any real number can never be negative; therefore, the square root of a negative number is not defined. But to solve the question we will first express the given square root \[ - 72\] in the form of multiple of two square roots and then solve both of them one by one and reach the desired result.

Complete step-by-step solution:
Given, \[\sqrt { - 72} \]
Using a property of square root that it can be written in the form of multiple of two square roots,
\[\sqrt { - 72} = \sqrt {72} \times \sqrt { - 1} \;\;\;\;\;\; \ldots \left( 1 \right)\]
We will solve \[\sqrt {72} \] at first,
\[\sqrt {72} = \sqrt {6 \times 6 \times 2} \]
Taking squares outside the root, we get
\[\sqrt {72} = 6\sqrt 2 \]
Now, equation \[\left( 1 \right)\] becomes,
\[\sqrt { - 72} = 6\sqrt 2 \times \sqrt { - 1} \]
Now, we know squares of any real number can never be negative. Hence \[\sqrt { - 1} \] is not a real number but a complex number and it is denoted by the symbol \[i\].
Therefore, \[\sqrt { - 72} = 6\sqrt 2 i\]
Additional information: Number system has only two types of numbers i.e. real numbers and complex numbers. Generally, we use complex numbers to find the roots of the quadratic equation when \[{b^2} - 4ac < 0\]

Note: We have studied only real numbers so far. We have not seen anything like a negative number inside the square root. But, here a new concept is getting introduced, Complex numbers. Complex numbers are the numbers which can be expressed in the form of \[a + ib\], where \[a,b\] are the real numbers and \[{i^2} = - 1\]. In other words, we can say that the complex numbers are the numbers which are not real numbers.