Question

How do you find the square root of 54?

Answer 1

Hint: In the above question, square root can be calculated in such a way that a number when multiplied by itself gives the original number present in the question. Basically, we need to find factors of 54 and further reduce the factors into a single value by grouping the pair of common numbers in it and writing it as one number.

Complete step by step answer:
First, we need to check whether the number given is a perfect square number or not.
Perfect square means a square of a number which results in giving that number. But 54 is not a perfect square since it doesn’t reduce to a single integer number.
So, now reduce 54 to its factors. By using the Distributive property of square roots, we need to split the root of 54. For example, it can be written generally as
\[ \Rightarrow \sqrt {ab} = \sqrt a \sqrt b \]
54 has factors 3,3,3,2.
\[ \Rightarrow 54 = 3 \times 3 \times 3 \times 2\]
Taking square root on both sides,
\[ \Rightarrow \sqrt {54} = \sqrt {3 \times 3 \times 3 \times 2} \]
Now by grouping the pair of same numbers into one, the number is taken out from the square root.
\[ \Rightarrow \sqrt {54} = \sqrt {3 \times 3 \times 3 \times 2} \]
\[
\Rightarrow \sqrt {54} = 3\sqrt {3 \times 2} \\
\Rightarrow \sqrt {54} = 3\sqrt 6 \\
 \]
 Every number will have real square roots positive and negative. Here we are considering only non-negative square roots. Therefore, the square root of 54 is \[3\sqrt 6 \].

Note: An alternative and easy way to solve this by writing two factors of 54 under the square root sign and if the factor is already in square, we can take that number has common and write it outside the square root i.e.,\[\sqrt {54} = \sqrt {9 \times 6} = \sqrt {{{(3)}^2} \times 6} = 3\sqrt 6 \].