Question

What is the square root of $\dfrac{9}{169}$ ?

Answer 1

Hint: Here, we need to square root a fraction. We do so by square rooting both the numerator and the denominator. We perform prime factorisation of the two individually and then square root them. Then, we again express them as a fraction.

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 $9$ and $169$ gives,
\[\begin{align}
  & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & ~~~\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]

\[\begin{align}
  & 13\left| \!{\underline {\,
  169 \,}} \right. \\
 & 13\left| \!{\underline {\,
  13 \,}} \right. \\
 & ~~~\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Which can be written as
\[\begin{align}
  & 3\times 3={{3}^{2}} \\
 & 13\times 13={{13}^{2}} \\
\end{align}\] .
After square rooting, clearly it gives $3$ and $13$ .
Thus, we can conclude that the square root of $\dfrac{9}{169}$ is $\dfrac{3}{13}$.

Note: The problem here is pretty easy as the numbers are perfect squares. But the case becomes a little more calculative when the numbers involved are not perfect squares. We should do that carefully. Both 9 and 169 are perfect squares, so if we learn the perfect square of numbers till 20, it will be helpful in solving questions easily.