Question

What is the answer to 2 times the square root of 18?

Answer 1

Hint: We must first find the prime factorisation of the number inside the radical sign, and then look for the prime numbers that are appearing at least twice. We can then bring such prime numbers outside the radical sign to get the answer to this verbal expression.

Complete step-by-step answer:
We know that a square root is the smallest radical. We also know very well that any expression or number, which is inside the root symbol is called a radicand. And we must also note that the small number, written in front of the root symbol, is called the index of the radical.
We are aware that the accepted form of representing a radical is in simplified radical form, in which the number inside the radical symbol does not contain any square, or cube of any number, as per the index of that radical.
Also, for square roots, we have index = 2.
In this problem, we have to express ‘2 times the square root of 18’ in simplified radical form.
First, let us find the prime factorization of 18.
$\begin{align}
  & \text{2}\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Hence, we have $18=2\times 3\times 3$ .
We can see that, in the prime factorisation, 2 is occurring just once and 3 is occurring twice.
Thus, we can very easily say that $\sqrt{18}=3\sqrt{2}$.
Hence, ‘2 times the square root of 18’ is $2\times 3\sqrt{2}=6\sqrt{2}$.
Thus, the answer to 2 times the square root of 18 is $6\sqrt{2}$.

Note: We must be very clear with the terminologies used in context of the radicals. We must also keep in mind that the prime factorization of each number is unique. Thus, it is totally fine to comment about the number by analysing its prime factorization.