Question

How do you simplify $\sqrt {\dfrac{{500}}{{720}}} $?

Answer 1

Hint:According to the given question, we have to simplify $\sqrt {\dfrac{{500}}{{720}}} $.
So, first of all we have to simplify both the numerator and denominator of the given question by prime factorisation as mentioned below.
Prime factorisation: In this method, we have to divide the number by the first prime number 2 and continue dividing by 2 until we get a decimal or remainder. Then divide by 3, 5, 7 etc. until the only numbers left are prime numbers.
For example: The prime factorisation of 228 as given below,
$ \Rightarrow 228 = 2 \times 2 \times 3 \times 19$
Now, we have to come out with numbers from the square root which comes 2 times and the remaining term is solved by dividing.

Complete step by step answer:
Step 1: First of all we have to simplify both the numerator and denominator of the given question by prime factorisation as mentioned in the solution hint.
The prime factorisation of 500$ = 2 \times 2 \times 5 \times 5 \times 5$
The prime factorisation of 720$ = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5$
Step 2: Now, we have to come out with the numbers from the square root of the numbers that are mentioned in the solution step 1, which comes 2 times.
$
   \Rightarrow \sqrt {500} = \sqrt {2 \times 2 \times 5 \times 5 \times 5} \\
   \Rightarrow 2 \times 5 \times \sqrt 5 \\
 $
$
   \Rightarrow \sqrt {720} = \sqrt {2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5} \\
   \Rightarrow 2 \times 2 \times 3 \times \sqrt 5 \\
 $
Step 3: Now, we have to divide both square roots as mentioned in the solution step 2.
$ \Rightarrow \dfrac{{\sqrt {500} }}{{\sqrt {720} }} = \dfrac{{2 \times 5 \times \sqrt 5 }}{{2 \times 2 \times 3 \times \sqrt 5 }}$
Now, we have to solve the above expression by canceling the common term in numerator and denominator both.
$ \Rightarrow \sqrt {\dfrac{{500}}{{720}}} = \dfrac{5}{6}$
Final solution: Hence, the simplified value of$\sqrt {\dfrac{{500}}{{720}}} $is$\dfrac{5}{6}$.

Note:
It is necessary to simplify both the numerator and denominator of the given question by prime factorisation as mentioned in the solution hint.
It is known that to take out numbers from the square root which comes 2 times.