Question

What is the square root of zero?

Answer 1

Hint: Before solving this problem, we should have a clear understanding of indices and roots. We should know what square rooting means. We will solve square rooting by doing the prime factorisation of the original number. But, as zero multiplication gives zero itself, so the square root of zero will also be zero.

Complete step-by-step solution:
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 $196$ gives,
\[\begin{align}
  & 2\left| \!{\underline {\,
  196 \,}} \right. \\
 & 2\left| \!{\underline {\,
  98 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & ~~~\left| \!{\underline {\,
  7 \,}} \right. \\
\end{align}\]
Which can be written as $2\times 2\times 7\times 7={{2}^{2}}\times {{7}^{2}}$ . After square rooting, clearly it gives $2\times 7$ . But, zero can be written as $0=0\times 0$ . So, its square root will also give $0$ .
Therefore, we can conclude that the square root of zero is $0$.

Note: Square rooting problems are quite simple and so, students should not make a mistake here. Square rooting can be done by prime factorisation or by using a calculator. But, while evaluating the square root of zero, we often consider it as one of the indeterminate forms. But this is not the case.