Question

Find the square root of the following numbers by the method of prime factorisation.
(i). 121
(ii) 225
(iii) 441
(iv) 484
(v) 676
(vi) 900
(vii) 1296
(viii) 1024

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.

Complete step by step answer:
In this question, we are given some numbers and we have to find the square root of each of them by using the prime factorization method. We will follow certain steps to find the square root by prime factorization as follows,
1. We have to first divide the given number into its prime factors.
2. We have to now pair similar factors that are equal to each other.
3.Now, we will take one factor from each of the pairs.
4. We will then take the product of the factors thus obtained in step 3.
For example, we will find the square root of 256. So, we will factorise it first as given below,
$\begin{align}
  & 2\left| \!{\underline {\,
  256 \,}} \right. \\
 & 2\left| \!{\underline {\,
  128 \,}} \right. \\
 & 2\left| \!{\underline {\,
  64 \,}} \right. \\
 & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & \text{ }2 \\
\end{align}$
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 the pairs of these factors, so 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 of this as 16. Therefore, the square root of 256 is 16. Similarly, we will now find the square roots of all the given numbers one by one.
(i) 121
We will factorise it first as below,
$\begin{align}
  & 11\left| \!{\underline {\,
  121 \,}} \right. \\
 & \text{ }11 \\
\end{align}$
So, we write the factors as $11\times 11$. And we will get the pair as $\overline{11\times 11}$. Now we take one factor of the pair , that is 11.
So, we get the square root of 121 as 11.
(ii) 225
We will first factorise it as below,
$\begin{align}
  & 3\left| \!{\underline {\,
  225 \,}} \right. \\
 & 3\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & \text{ 5} \\
\end{align}$
So, the factors are, $3\times 3\times 5\times 5$. Pairing them, we will get, $\overline{3\times 3}\times \overline{5\times 5}$. So, taking one factor out of each pair, we get, $3\times 5$. Now we will find their product, which is 15.
So, the square root of 225 is 15.
(iii) 441
We will first factorise it as follows,
$\begin{align}
  & 3\left| \!{\underline {\,
  441 \,}} \right. \\
 & 3\left| \!{\underline {\,
  147 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & \text{ 7} \\
\end{align}$
So, the factors of 441 are $3\times 3\times 7\times 7$. Now we can get the pairs as $\overline{3\times 3}\times \overline{7\times 7}$. And taking one factor from each of the pairs we get, $3\times 7$. The product of $3\times 7$ is 21.
So, the square root of 441 will be 21.
(iv) 484
So, we will factorise it as follows,
 $\begin{align}
  & 2\left| \!{\underline {\,
  484 \,}} \right. \\
 & 2\left| \!{\underline {\,
  242 \,}} \right. \\
 & 11\left| \!{\underline {\,
  121 \,}} \right. \\
 & \text{ 11} \\
\end{align}$
So, we get the factors as, $2\times 2\times 11\times 11$. We will pair them as, $\overline{2\times 2}\times \overline{11\times 11}$. We will take 2 and 11 from this pair and get their product as 22.
So, the square root of 484 will be 22.
(v) 676
We factorise it as,
$\begin{align}
  & 2\left| \!{\underline {\,
  676 \,}} \right. \\
 & 2\left| \!{\underline {\,
  338 \,}} \right. \\
 & 13\left| \!{\underline {\,
  169 \,}} \right. \\
 & \text{ 13} \\
\end{align}$
We get the factors of 676 as, $2\times 2\times 13\times 13$. By pairing them, we get $\overline{2\times 2}\times \overline{13\times 13}$. So, taking one each from the pair, we get $2\times 13$ and their product is 26.
So, the square root of 676 will be 26.
(vi) 900
We will factorise it as follows,
$\begin{align}
  & 2\left| \!{\underline {\,
  900 \,}} \right. \\
 & 2\left| \!{\underline {\,
  450 \,}} \right. \\
 & 3\left| \!{\underline {\,
  225 \,}} \right. \\
 & 3\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & \text{ 5 } \\
\end{align}$
So, the factors are, $2\times 2\times 3\times 3\times 5\times 5$, and their pairs are, $\overline{2\times 2}\times \overline{3\times 3}\times \overline{5\times 5}$. Taking one from each of the pairs, we get, $2\times 3\times 5$. Their product is 30.
So, the square root of 900 is 30.
(vii) 1296
We will factorise 1296 as,
$\begin{align}
  & 2\left| \!{\underline {\,
  1296 \,}} \right. \\
 & 2\left| \!{\underline {\,
  648 \,}} \right. \\
 & 2\left| \!{\underline {\,
  324 \,}} \right. \\
 & 2\left| \!{\underline {\,
  162 \,}} \right. \\
 & 3\left| \!{\underline {\,
  81 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & \text{ 3 } \\
\end{align}$
So, the factors are, $2\times 2\times 2\times 2\times 3\times 3\times 3\times 3$ and we can pair them as, $\overline{2\times 2}\times \overline{2\times 2}\times \overline{3\times 3}\times \overline{3\times 3}$. Now we will take one factor each from the pairs, so we get, $2\times 2\times 3\times 3$. We get their product as, 36.
So, the square root of 1296 is 36.
(viii) 1024
We will find the factors as follows,
$\begin{align}
  & 2\left| \!{\underline {\,
  1024 \,}} \right. \\
 & 2\left| \!{\underline {\,
  512 \,}} \right. \\
 & 2\left| \!{\underline {\,
  256 \,}} \right. \\
 & 2\left| \!{\underline {\,
  128 \,}} \right. \\
 & 2\left| \!{\underline {\,
  64 \,}} \right. \\
 & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & \text{ 2 } \\
\end{align}$
So, the factors are, $2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$. We can pair them as follows, $\overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}$. We now take one factor out of the pairs, so we get, $2\times 2\times 2\times 2\times 2$ and their product is 32.
So, the square root of 1024 will be 32.

Note:
The students should be careful to choose similar factors while choosing from the pairs, otherwise, they may end up with an incorrect answer. This method is used for smaller numbers and for large numbers the square root is found by the division method.