Question

How do you do square root of $98$ divided by square root of $18$?

Answer 1

In this question, we are given a statement “square root of $98$ divided by square root of $18$”.

Firstly, we will convert the statement into mathematical form. Afterwards, we will attempt to factorize the terms in numerator and denominator to simplify the expression. We will then look for some common factors (if any) to cancel out, and then we will perform a simple division to obtain the answer.

Complete step by step solution:
(i)
We are given to calculate “square root of $98$ divided by square root of $18$”
Since, we know that square root of $98$ can be written as $\sqrt {98}$ and square root of $18$ can be written as $\sqrt {18}$ and we have to divide $\sqrt {98}$ by $\sqrt {18}$. Therefore, we have to calculate:

$\dfrac{{\sqrt {98} }}{{\sqrt {18} }}$

(ii)
Now, we will simplify the given division by factoring the numerator and denominator separately and cancelling out the common factors.

So, as we know that the prime factorization of $98$ is:
$98 = 2 \times 7 \times 7$

Square rooting both the sides, we will get:
$\sqrt {98} = \sqrt {2 \times 7 \times 7}$

Which can also be written as:
$\sqrt {98} = \sqrt {2 \times {7^2}}$

Now, as we know that $\sqrt {ab} = \sqrt a \times \sqrt b$, we can write $\sqrt {98}$ as:
$\sqrt {98} = \sqrt 2 \times \sqrt {{7^2}}$

Since, we also know that square root of a square of a number is the number itself i.e., $\sqrt {{a^2}} = a$.

Therefore, we can write the above equation as:
$\sqrt {98} = \sqrt 2 \times 7$

(iii)
Now, we will see the prime factorization of $18$, i.e.,
$18 = 2 \times 3 \times 3$

Square rooting both the sides, we will get:
$\sqrt {18} = \sqrt {2 \times 3 \times 3}$

Which can also be written as:
$\sqrt {18} = \sqrt {2 \times {3^2}}$

As we know that $\sqrt {ab} = \sqrt a \times \sqrt b$, we can write $\sqrt {18}$ as:

$\sqrt {18} = \sqrt 2 \times \sqrt {{3^2}}$

Since, we also know that the square root of a square of a number is the number itself i.e., $\sqrt {{a^2}} = a$.

Therefore, we can write the above equation as:
$\sqrt {18} = \sqrt 2 \times 3$

(iv)
Now, as we have got the simplified and factored version of the numerator and the denominator, we will substitute their value in the original expression i.e., $\dfrac{{\sqrt {98} }}{{\sqrt {18} }}$.

Therefore, we will get:
$\dfrac{{\sqrt {98} }}{{\sqrt {18} }} = \dfrac{{\sqrt 2 \times 7}}{{\sqrt 2 \times 3}}$

As we can see that $\sqrt 2$ is common in numerator as well as denominator, it can be cancelled out.

Therefore, we will get:
$\dfrac{{\sqrt {98} }}{{\sqrt {18} }} = \dfrac{7}{3}$

Dividing $7$ by $3$, we will get:
$\dfrac{{\sqrt {98} }}{{\sqrt {18} }} = 2.333$

Hence, square root of $98$ divided by square root of $18$ is $2.333$

Note: There is an alternate method to solve this question. Since, we know that $\dfrac{{\sqrt a}}{{\sqrt b }} = \sqrt {\dfrac{a}{b}}$, we could also write $\dfrac{{\sqrt {98} }}{{\sqrt {18} }}$ as $\sqrt {\dfrac{{98}}{{18}}}$.After this, we could have factorized the numerator and denominator separately inside the square root and cancelled the common term out (which would have been $2$) and when we get a simple division of two numbers, we could have divided them and then calculated their square root.