Question

What is the square root of $18$ plus the square root of $72$ ?

Answer 1

Hint: The first thing that we need to do is to convert the words into mathematical expressions. It becomes $\sqrt{18}+\sqrt{72}$ . We then perform the prime factorisation of the numbers $18,72$ . Having done that, we can now easily find out the square root of them. Adding the square roots, we get the answer.

Complete step-by-step solution:
First, we need to convert the words into an equation form. Plus means “addition”, so we add $\sqrt{18}$ and $\sqrt{72}$ . But, before that, we need to square root the numbers. 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 $18$ and $72$ gives,
\[\begin{align}
  & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & ~~~\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]

\[\begin{align}
  & 2\left| \!{\underline {\,
  72 \,}} \right. \\
 & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & ~~\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Which can be written as $2\times 3\times 3={{3}^{2}}\times 2$ and $2\times 2\times 2\times 3\times 3={{2}^{2}}\times 2\times {{3}^{2}}$ .
After square rooting, clearly it gives $3\sqrt{2}$ and $6\sqrt{2}$ . The addition of the two leads to $3\sqrt{2}+6\sqrt{2}$ .
The term $\sqrt{2}$ can be taken from each of the two terms. After taking it common, the expression becomes,
$\Rightarrow \left( 3+6 \right)\sqrt{2}=9\sqrt{2}$ .
Thus, we can conclude that the square root of $18$ plus the square root of $72$ is $9\sqrt{2}$.

Note: In this problem, we must carefully understand each and every word, else our word to arithmetic conversion would get wrong. We can also solve the problem using another way. We use a calculator and then calculate each of the square roots and then add them.