Question

What is the square root of $147?$

Answer 1

Hint: We will write the given number as a product of its factors in order to find the square root of the given number. We know the identity saying that $\sqrt{ab}=\sqrt{a}\sqrt{b}.$ The given number is a composite number.

Complete step by step solution:
We are asked to find the square root of the number $147.$
We know that the number $147$ is not a prime number. That is, the given number is a composite number. Therefore, this number has factors that are not $1$ and itself.
So, we can write this number as a product of prime numbers.
Thus, we will get $147=3\times 7\times 7.$
We can write this again as $147=3\times 49.$
We will get $\sqrt{147}=\sqrt{3\times 49}.$
Let us use the identity given by $\sqrt{ab}=\sqrt{a}\sqrt{b}.$
We will get $\sqrt{147}=\sqrt{3}\sqrt{49}.$
We know that the square root of $49$ is equal to $7.$
Therefore, we will get $\sqrt{49}=7.$
So, we will get $\sqrt{147}=7\sqrt{3}.$
Therefore, if we write it as a statement, we will get, the square root of the given number $147$ is $7$ times the square root of $3.$
Hence the square root of the given number is equal to $\sqrt{147}=7\sqrt{3}.$

Note: The identity we have used in the given problem can be generalized for a number of terms. We will get $\sqrt{{{a}_{1}}\cdot {{a}_{2}}\cdot {{a}_{3}}\cdot ...\cdot {{a}_{n}}}=\sqrt{{{a}_{1}}}\sqrt{{{a}_{2}}}\sqrt{{{a}_{3}}}...\sqrt{{{a}_{n}}}.$ A number that can be written as a product of numbers other than $1$ and itself is called a composite number. A number that cannot be written as a product of numbers other than $1$ and itself is called a prime number. Remember that the composite numbers can be written as a product of prime numbers.