Question

Using the prime factorization method, find the following number is a perfect square: 1176.

Answer 1

Hint: We will first factorize the given numbers into their prime factors and then pair similar factors. Then we will choose one for each pair and at last, take their product to get the answer. And thus we will get our required answer.

Complete step by step answer:
In this question we are given a number and we have to find the square root of that by using the prime factorization method. We will follow certain steps to find the square root by prime factorization as follows.
We have to first divide the number into prime factors.
We have to now pair similar factors that are equal to each other.
Now, we will take one factor from each of the pairs.
We will then take the product of the factor thus obtained in step 3.
For example we will find the square root of 256.
So, we will factorize it first as given below.
\[\begin{align}
  & 2\left| \!{\underline {\,
  256 \,}} \right. \\
 & 2\left| \!{\underline {\,
  128 \,}} \right. \\
 & 2\left| \!{\underline {\,
  64 \,}} \right. \\
 & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, we can write 256 as \[2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\].
Now we will make pairs of these factors, we get \[\overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\]. Now, we will take a single factor from each of these pairs. So, we get \[2\times 2\times 2\times 2\]. So, we get the product as 16.
Similarly, for 1176, we will find the square root, then
\[\begin{align}
  & 2\left| \!{\underline {\,
  1176 \,}} \right. \\
 & 2\left| \!{\underline {\,
  588 \,}} \right. \\
 & 2\left| \!{\underline {\,
  294 \,}} \right. \\
 & 3\left| \!{\underline {\,
  147 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, we get factors as \[2\times 2\times 2\times 3\times 7\times 7\]. We will pair them as \[\overline{2\times 2}\times 2\times 3\times \overline{7\times 7}\]. So, here only two pairs and two are single (unpaired). So, we will take this as \[2\times 7\times \sqrt{2\times 3}\] which will be as \[14\sqrt{6}\] and if we solve this then we got 34.292. So, the square root of 1176 is 34.292.

Note: While solving this type of questions you have to mind that if all pairs are made correctly then make them and if any digits are remaining then take them as a square root value and then produce them.