Question

How do you simplify \[\sqrt{84}?\]

Answer 1

Hint: We are given \[\sqrt{84},\] we have to simplify it. We will learn how to simplify such types of expressions with some examples to get the understanding and then we will work on our question that is \[\sqrt{84}.\] Then we will take the pairs out of the root leaving all unpaired inside it and then we will simplify and solve.

Complete answer:
We are given an equation, a term with the square root, inside it we have 84. To learn how to simplify such questions, we will learn some examples related to this to solve such problems where we have radical \[\left( \sqrt{{}} \right)\] in the numerator only and the denominator is simply one. This type of question is mainly based on prime factorization to simplify. Like, we will consider we have \[\sqrt{8}.\] So, we have 8 inside the square root, we will factor 8 into its prime factor. So, we get,
\[8=2\times 2\times 2\]
So, we get,
\[\sqrt{8}=\sqrt{2\times 2\times 2}\]
Now, properties of the square root are only a pair of terms that can be taken out of it as we can have 3 pieces of 2. So, 2 pieces of 2 will make one pair. So, it will come out and the remaining one will stay inside the root. So, we can write,
\[\Rightarrow \sqrt{8}=2\times \sqrt{2}\]
\[\Rightarrow \sqrt{8}=2\sqrt{2}\]
Similarly, say we have \[\sqrt{16}.\] So,
\[\sqrt{16}=\sqrt{2\times 2\times 2\times 2}\]
Here, we are getting paired up. So, all terms will come out and there will be nothing inside the root.
\[\Rightarrow \sqrt{16}=2\times 2\]
\[\Rightarrow \sqrt{16}=4\]
So, the simplification of \[\sqrt{16}\] is 4.
Now, we will work on our problem, we have \[\sqrt{84}\] so we will start our solution by finding the prime factor of 84. So, we get,
\[84=2\times 2\times 7\times 3\]
We can see here that in the prime factorization of 84, only 2 is making its pair while 3 and 7 are single digits. So, only 2 will come out and 3 and 7 stay inside the root. So, we get,
\[\Rightarrow \sqrt{84}=\sqrt{2\times 2\times 3\times 7}\]
\[\Rightarrow \sqrt{84}=2\times \sqrt{3\times 7}\]
As, \[3\times 7=21,\] so we get,
\[\Rightarrow \sqrt{84}=2\sqrt{21}\]

Note: Students need to note that outside the square root only one term we write, means a pair of 2 becomes one when taken out of the square root. So, we need to be careful about this. Also, we need to form the prime factor and nothing less than that. If we do not make a prime factor, our solution will not be correct.