Question

Find the number whose square is the given number 1156.

Answer 1

Hint: First find a prime number which divides the given number. Now write this number as a product of the prime number and respective quotient. Repeat this step till you get 1. Simply saying do prime factorization of this number. Now try to pair to some factor. If you do these steps then check whether any prime factor is left above if not, we can say it is a perfect square. Now for every pair of prime numbers combine the pair and write it as a single number. Now multiplying all of them you get the required result.

Complete step-by-step answer:
Given the number in the question, and asked for the value to find: 1156.
We should find a number whose square is 1156. Let us assume the required number to be variable x. by given condition in the question, we can write equation:
${{x}^{2}}=1156$
By applying square root on both sides of equation, we get:
$x=\sqrt{1156}...............(i)$
So, we need the square root of number 1156.
First, we need to prove that 1156 is a perfect square.
Perfect square: A number whose square root is an integer is called a perfect square.
To prove this, we need to apply prime factorization to 1156.
Prime factorization: The process of finding all the prime numbers which divide this number and writing this in terms of the product of them is called prime factorization of a number.
By dividing the number with 2, we get a quotient:
$1156=2\times 578$
By dividing the number 578 by 2, we get quotient:
$1156=2\times 2\times 289$
We know that 289 can be written as $17\times 17$, we get:
$1156=2\times 2\times 17\times 17$
So, we can pair these up as $\left( 2,2 \right);\left( 17,17 \right)$, so we proved that 1156 is a perfect square.
By above equation, we can write the number 1156 as:
$1156={{2}^{2}}\times {{17}^{2}}$
By applying square root on both sides of equation, we get:
$\sqrt{1156}=\sqrt{{{2}^{2}}\times {{17}^{2}}}$
By simplifying, we get the equation as:
$\sqrt{1156}=34$
Because: $\sqrt{{{a}^{2}}\times {{b}^{2}}}={{\left( {{a}^{2}}{{b}^{2}} \right)}^{1/2}}=a\times b$
By substituting equation (i) back into this equation, we get x=34.
Therefore, the required result is 34.

Note: Be careful while applying the prime factorization. Generally, you must do till you get 1. But here we get a square of prime (289). So, we stopped at that point. Generally, students use division methods. It is also correct but a long process.