Question

How do you simplify $\sqrt {128} $ ?

Answer 1

Hint:
For finding the square root of $128$ we need to break down this number into the product of numbers by doing the prime factor. Then we will solve this by making a pair if it will be otherwise we will write the same as we got from the factor.

Complete Step by Step Solution:
So the prime breakdown of the number $128$ will be
$ \Rightarrow 128 = 2 \times 64$
Again doing the breakdown of $64$ , we will get
$ \Rightarrow 128 = 2 \times 2 \times 2 \times 16$
Again doing the breakdown of $16$ , we will get
$ \Rightarrow 128 = 2 \times 2 \times 2 \times 2 \times 8$
Similarly, we will do this breakdown till possible and finally, we will get the number
$ \Rightarrow 128 = {2^7}$
So, now from the question, we can write it as
$ \Rightarrow \sqrt {128} = \sqrt {{2^7}} $
The right side of the equation can be written as
$ \Rightarrow \sqrt {128} = \sqrt {{2^6} \times {2^2}} $
Again the right side of the equation can be written as
$ \Rightarrow \sqrt {128} = {2^3}\sqrt 2 $
And on solving the above equation, we get
$ \Rightarrow \sqrt {128} = 8\sqrt 2 $

And therefore, the square root of $128$ will be $8\sqrt 2 $.

Note:
This can be solved by using the shorter way too. Only we need to break it into such a way that there should be a square of it. Like we can write the above question as
$ \Rightarrow \sqrt {128} = \sqrt {64 \times 2} $
And taking the RHS, we will split both the root so we get
$ \Rightarrow \sqrt {128} = \sqrt {64} \times \sqrt 2 $
And as we know that $\sqrt {64} = 8$ , hence by substituting this value, we will get the equation as
$ \Rightarrow \sqrt {128} = 8\sqrt 2 $
So, in this way also we can solve such types of questions.