Question

How do you simplify $\sqrt {45{x^2}} $?

Answer 1

Hint: We will first prime factorize 45 and then write the square of x as multiples of two x’ s. Now, to get the square root, we make pairs of like terms and bring one out among them.

Complete step-by-step answer:
We are given that we are required to find $\sqrt {45{x^2}} $.
In this, there is a 45, let us first find its prime factorization.
$ \Rightarrow 45 = 3 \times 3 \times 5$
We can clearly see that 3 is making a pair here.
Now, we have the following:-
$ \Rightarrow 45{x^2} = 3 \times 3 \times 5 \times x \times x$
Taking the square root on both the sides, we will get:-
$ \Rightarrow \sqrt {45{x^2}} = \sqrt {3 \times 3 \times 5 \times x \times x} $
Since, 3 and x are in pair in the square root in right hand side, we can bring out one of that pair out and get the following expression:-
$ \Rightarrow \sqrt {45{x^2}} = 3x\sqrt 5 $

Hence, the required answer is $3x\sqrt 5 $.

Note:
The students must note that it is very important to prime factorize the number so that we can get the maximum out of the square root as much as possible. If we would have had one more 5 inside the square root, then we would have got the rational number and the square root would have been eradicated. But, right now, we have a square root of 5, so we have an irrational number as the answer because the rational number multiplied by irrational number is irrational unless the rational we used is 0. So, if x is not equal to 0 in the given question, we have an irrational answer.
The students must know that prime factorization of a number means that we break that number into multiple factors in their possible smallest prime factors and thus we have the prime factorization. In 45, the multiples came out to be 5 and 9 in the first place and then we broke 9 again into 3 times 3 and thus we got 3, 3 and 5 in its prime factorization.