Question

What is the square root of 550?

Answer 1

Hint: We know that to find the square root of any number, we must, first, find the prime factorisation of that number, and then look for those prime numbers that are appearing at least twice. We can bring such prime numbers outside the square root sign to get the square root.

Complete step-by-step answer:
In this problem, we need to find the square root of 550, that is, if $y=\sqrt{550}$, we need to value of $y.$
First of all, we need to find the prime factorization of 550.
$\begin{align}
  & \text{ 2}\left| \!{\underline {\,
  550 \,}} \right. \\
 & \text{ }5\left| \!{\underline {\,
  275 \,}} \right. \\
 & \text{ 5}\left| \!{\underline {\,
  55 \,}} \right. \\
 & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Hence, we have $550=2\times 5\times 5\times 11$ .
Here, we can see that, in the prime factorisation, 2 is present once, 5 is present twice, and 11 is present once.
Thus, $\sqrt{550}=\sqrt{2\times 5\times 5\times 11}$
We know that we can bring those prime factors out of the square root that is occurring more than once in the prime factorization.
So, from this prime factorisation, we can see that bring the factor 5 outside the square root.
So, we now have, $\sqrt{550}=5\sqrt{2\times 11}$
We can see that there are no repeated primes inside the square root. Thus, we can’t reduce this any further.
So, $\sqrt{550}=5\sqrt{22}$
Since, the final answer includes a square root, we can say that 550 is not a perfect square.
Hence, the square root of 550 is $5\sqrt{22}$.

Note: We must keep in mind that the prime factorisation of each number is unique. So, it is totally fine to comment about the number by analysing its prime factorization. We should be aware that square root is denoted both by $\sqrt{{}}$ and $\sqrt[2]{{}}$ .