Question

How do you simplify \[\sqrt{972}\]?

Answer 1

Hint: The symbol \[\sqrt{{}}\] is known as the radical sign. In this question, we have to find the value of \[\sqrt{972}\] that means we have to find the simplest radical form 972 For finding that value, first we will find the prime factorization of 972. We will check which factors are making the perfect square so that we will find the square root or radical form of 972 easily.

Complete step by step answer:
Let us solve the question.
Let us find out the factorization of 972.
\[\begin{align}
  & 2\left| \!{\underline {\,
  972 \,}} \right. \\
 & 2\left| \!{\underline {\,
  486 \,}} \right. \\
 & 3\left| \!{\underline {\,
  243 \,}} \right. \\
 & 3\left| \!{\underline {\,
  81 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
In the above factorization, the first step is that we have taken the smallest prime factor of 972 or the smallest number which is divisible by 972 except 1. From there, we get a number 486 by dividing 2 by the number 972. After that, we will similarly find the smallest number which is divisible by 486 and write that number on the left side of column to make a new number. That number will be \[\dfrac{486}{2}=243\]. Similarly, for 243, the smallest number which is a prime factor of 243 is 3. Then, we get \[\dfrac{243}{3}=81\]. We will continue the process until we don’t get 1. When we are at the last step, then there will be no prime factors of 1. Then, we will stop here.
Now, we get that the prime factorization of 972 is \[2\times 2\times 3\times 3\times 3\times 3\times 3\].
Now, we will check how many are making squares in that factorization so that we will get a perfect square root.
\[972=2\times 2\times 3\times 3\times 3\times 3\times 3\times 1={{2}^{2}}\times {{3}^{2}}\times {{3}^{2}}\times 3\]
Here, we can see that there are one 2 and two 3s which are making squares.
Therefore,
\[\sqrt{972}=\sqrt{{{2}^{2}}\times {{3}^{2}}\times {{3}^{2}}\times 3}=\sqrt{{{2}^{2}}}\times \sqrt{{{3}^{2}}}\times \sqrt{{{3}^{2}}}\times \sqrt{3}=2\times 3\times 3\times \sqrt{3}=18\times \sqrt{3}\]

Hence, \[\sqrt{972}=18\sqrt{3}\].

Note: For solving this question, we should know how to find the multiplication factors or prime factorization of a number. And also what the prime factors are. The prime factor of a number is the factor (where factors are the numbers we multiply them to get a new or desired number) of a number which is a prime.