Question

What is the square root of negative $8$ ?

Answer 1

Hint: Before solving this problem, we should have a clear understanding of roots and complex numbers. We should know what the square root of negative one means. We will solve the square root by doing the prime factorisation of the original number i.e $8$. Then, we will use the fact that $\sqrt{-1}$ is $i$. Multiplying the square root of $8$ and square root of $-1$ will give us the answer.

Complete step by step answer:
Indices are a representation of repetitive multiplication of the same kind. Repetitive multiplication if shown in the form $a\times a\times a\times a\times ....$ will become tedious and time taking. So, in order to avoid it, we have implemented the indices representation. Here, we represent by the original number with the number of multiplications written as a superscript. For example, three times multiplication of two will be $2\times 2\times 2$ which can be written as ${{2}^{3}}$ . Indices can be called as an operation of numbers. The inverse operation of indices is called square rooting, cube rooting and so on. Square rooting means to break down a number into two other similar numbers such that their product gives the original number. For example, the square root of $4$ gives $2$ since $2\times 2$ implies $4$ .
In square rooting, we use prime factorisation to break down a number into its prime factors. Prime factorisation gives the product of prime factors. For example, the prime factorisation of $8$ gives,
\[\begin{align}
  & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & ~~~\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Which can be written as $2\times 2\times 2={{2}^{3}}$ . After square rooting, clearly it gives $2\sqrt{2}$ .
By the given problem, $\sqrt{-8}$ can be written as $\sqrt{8}\times \sqrt{-1}$ by following the law of multiplication. Now, as we know that $\sqrt{-1}$ is called iota and is represented by “i”.

Therefore, we can conclude that the square root of $\sqrt{-8}$ is $2\sqrt{2}i$

Note: The first mistake that students make while solving these types of problems is that they ignore the negative sign. Doing so, it gives a completely different answer. The final answer should be cross-checked by squaring it and checking it with the original number.