Question

How do you simplify the square root of \[\dfrac{8}{{18}}\] ?

Answer 1

Hint: Here in this question, we have to simplify the given the number. The number is in the form of fraction. Now we have to apply the square root to the given number. The square root is applicable for the both numerator and denominator and then we are simplifying the given number.

Complete step-by-step answer:
The fraction is a whole number which contains the two parts numerator and denominator. The number presented above the line is a numerator and the number present the below line is denominator.
Now we have to simplify the given number by applying square root
 \[\sqrt {\dfrac{8}{{18}}} \] ----- (1)
If we see the number which is present in both numerator and denominator are not perfect squares. The perfect square is defined as when a number is multiplied twice then the resultant number is a perfect square.
So let us factor both numbers 8 and 18.

2	8
2	4
2	2
	1

Therefore the number 8 is written as \[2 \times 2 \times 2\]
Now let we factorise the number 18 we have

3	18
3	6
2	2
	1

Therefore the number 18 is written as \[3 \times 3 \times 2\]
The equation (1) is written as
 \[ \Rightarrow \sqrt {\dfrac{{2 \times 2 \times 2}}{{3 \times 3 \times 2}}} \]
In the numerator the number 2 is multiplied thrice we consider the number multiplied twice and write in the exponential form. In the denominator the number 3 is multiplied twice, we write in the form of exponential we have
 \[ \Rightarrow \sqrt {\dfrac{{{2^2} \times 2}}{{{3^2} \times 2}}} \]
By the square root property \[\sqrt {\dfrac{a}{b}} = \dfrac{{\sqrt a }}{{\sqrt b }}\] , using this property we have
 \[ \Rightarrow \dfrac{{\sqrt {{2^2} \times 2} }}{{\sqrt {{3^2} \times 2} }}\]
By the property of square root we have \[\sqrt {a \times b} = \sqrt a \times \sqrt b \] , using this the above equation is written as
 \[ \Rightarrow \dfrac{{\sqrt {{2^2}} \times \sqrt 2 }}{{\sqrt {{3^2}} \times \sqrt 2 }}\]
The square root and square cancels each other we have
 \[ \Rightarrow \dfrac{{2\sqrt 2 }}{{3\sqrt 2 }}\]
In the numerator and denominator we have \[\sqrt 2 \] so we cancel and we have
 \[ \Rightarrow \dfrac{2}{3}\]
Hence we have simplified.
So, the correct answer is “$\dfrac{2}{3}$”.

Note: When we want to find the square root of some number, let it be x. If x is a perfect square then we can obtain the result directly. Otherwise if x is not a perfect square let we factorise the x and if it possible we write the number in the form of exponential and then we simplify the number.