Question

How do you simplify $\sqrt{\dfrac{18}{5}}$?

Answer 1

Hint: First separate the square root for the numerator and the denominator as $\dfrac{\sqrt{18}}{\sqrt{5}}$. After finding the square root do rationalization if necessary. Multiply both the numerator and the denominator by $\sqrt{5}$ and do the necessary calculation to obtain the required solution.

Complete step by step solution:
The expression we have $\sqrt{\dfrac{18}{5}}$
As we know $\sqrt{\dfrac{a}{b}}$ can be written as $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$
So, $\sqrt{\dfrac{18}{5}}$ can be written as $\sqrt{\dfrac{18}{5}}=\dfrac{\sqrt{18}}{\sqrt{5}}$
To find the square root of a number we have to express it in terms of it’s prime factors. Then we have to take once from each pair of factors.
The prime factors of $18=3\times 3\times 2$
Simplifying the numerator, we get
$\sqrt{18}=\sqrt{3\times 3\times 2}=3\times \sqrt{2}=3\sqrt{2}$
Now, our expression becomes
$=\dfrac{3\sqrt{2}}{\sqrt{5}}$
Rationalization: It is a method by which we can write a fraction in such a way that the denominator contains only rational numbers.
To rationalize $\dfrac{3\sqrt{2}}{\sqrt{5}}$, we have to multiply both the numerator and the denominator with $\sqrt{5}$
Hence, multiplying both the numerator and the denominator by $\sqrt{5}$, we get
$=\dfrac{3\sqrt{2}\times \sqrt{5}}{\sqrt{5}\times \sqrt{5}}$
We know, $\sqrt{a}\times \sqrt{b}$ can be simplified using the product rule of radicals as $\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}$
So, $\sqrt{2}\times \sqrt{5}$ can be written as $\sqrt{2}\times \sqrt{5}=\sqrt{2\times 5}=\sqrt{10}$
And $\sqrt{5}\times \sqrt{5}$ can be written as $\sqrt{5}\times \sqrt{5}=\sqrt{5\times 5}=\sqrt{{{\left( 5 \right)}^{2}}}=5$
Combining the numerator and the denominator, we get
$=\dfrac{3\sqrt{10}}{5}$This is the required solution of the given question.

Note: Separation of square root for the numerator and the denominator should be the first approach for solving such questions. After finding the square root, rationalization must be done for further simplification.