Question

How do you find the square root of $ 1792 $ ?

Answer 1

Hint: Square root of a number is a value, which on multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as $ x = \sqrt y $ or we can express the same equation as $ {x^2} = y $ . Here we can see that $ 1792 $ is not a perfect square. To solve this we factorize the given number.

Complete step by step solution:
Given, $ \sqrt {1792} $
 $ 1792 $ can be factored as,
 $ 1792 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 $
We can see that $ 2 $ is multiplied eight times, so we write in exponential form and raise $ 2 $ to the power $ 8 $ . So, we get,
\[ \Rightarrow 1792 = {2^8} \times 7\]
Now, \[\sqrt {1792} = \sqrt {{2^8} \times 7} \]
Since we know that $ {2^8} $ is a perfect square. So, we can take this outside of the square root. Now, we know that the square root of $ {2^8} $ is $ {2^4} $ . So, we get,
So, \[\sqrt {1792} = {2^4} \times \sqrt 7 \]
Since $ 7 $ is not a perfect square, we have to leave it as it is. So, we multiply this and keep it inside the square root. We get,
 $ \Rightarrow \sqrt {1792} = {2^4}\sqrt 7 $
Now, we evaluate $ {2^4} $ to find the product. So, we get,
  $ \Rightarrow \sqrt {1792} = 16\sqrt 7 $
So, the square root of $ 1792 $ is $ 16\sqrt 7 $ .
So, the correct answer is “ $ 16\sqrt 7 $ ”.

Note: The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. To find the factors find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.