Question

What is the square root of $ 3 $ plus the square root of $ 27 $ ?

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 $ . In the given question, we are required to find the addition of two square roots given to us in the question. Also, we can see that $ 27 $ is not a perfect square. To solve this we factorize the given number.

Complete step-by-step answer:
Given, $ \sqrt {27} $
 $ 27 $ can be factorized as,
 $ 27 = 3 \times 3 \times 3 = {3^3} $
We can see that $ 3 $ is multiplied thrice, so we write in exponential form and raise $ 3 $ to the power $ 3 $ . So, we get,
 $ \Rightarrow 27 = {3^3} $
Now, $ \sqrt {27} = \sqrt {{3^3}} $
We know that \[{3^3} = {3^2} \times 3\]. So, $ {3^3} $ can be written as $ {3^2} \times 3 $ .
Now, $ \sqrt {27} = \sqrt {{3^2} \times 3} $
Since we know that $ {3^2} $ is a perfect square. So, we can take this outside of the square root we have,
So, $ \sqrt {27} = 3\sqrt 3 $
Since $ 3 $ is not perfect square, we can multiply this and keep it inside the square root,
 $ \Rightarrow \sqrt {27} = 3\sqrt 3 $
This is the simplified form of $ \sqrt {27} $ .
Now, we have to evaluate the sum of $ \sqrt 3 $ and $ \sqrt {27} $ .
 $ \sqrt {27} + \sqrt 3 = 3\sqrt 3 + \sqrt 3 $
 $ \Rightarrow \sqrt {27} + \sqrt 3 = 4\sqrt 3 $
Therefore, the sum of square root of $ 3 $ and the square root of $ 27 $ is $ 4\sqrt 3 $ .
So, the correct answer is “ $ 4\sqrt 3 $ ”.

Note: 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.