Question

How do you simplify square root \[\sqrt {180{x^2}} \] ?

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 \[{x^2} = y\] . Here we can see that 180 is not a perfect square but for any value of ‘x’ the given \[{x^2}\] is a perfect square. To solve this we factorize the given number.

Complete step-by-step answer:
We have, \[\sqrt {180{x^2}} \].
Here we have \[{x^2}\] . We know that square and square root will cancel out. That is \[\sqrt {{x^2}} = {x^{\dfrac{2}{2}}} = x\] .
 \[\sqrt {180{x^2}} = x\sqrt {180} \]
Now 180 can be factorized as,
 \[180 = 1 \times 2 \times 2 \times 3 \times 3 \times 5\] .
We can see that 2 and 3 is multiplied twice and multiplying this we get,
 \[180 = 4 \times 9 \times 5\]
Then,
 \[ \Rightarrow x\sqrt {180} = x\sqrt {4 \times 9 \times 5} \]
We know that 4 and 9 is a perfect square we can take it outside the radical symbol we get
 \[ = x \times2 \times 3\sqrt 5 \]
 \[ = 6x\sqrt 5 \] . This is the exact form we can stop here.
We know that \[\sqrt 5 = 2.236\] multiplying this with ‘6x’ we get
 \[ = 13.416x\] . This is the decimal form.
So, the correct answer is “13.416x”.

Note: Here \[\sqrt {} \] is the radical symbol used to represent the root of numbers. The number under the radical symbol is called radicand. 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.