Question

How do you simplify $\sqrt{\dfrac{16}{36}}$?

Answer 1

Hint: First take the square root for the numerator and the denominator separately. Then express both the numerator and the denominator in terms of their prime factors as both are square terms. Take each pair of factors only once from the radical for the required result.

Complete step by step solution:
The expression we have $\sqrt{\dfrac{16}{36}}$
As we know $\sqrt{\dfrac{a}{b}}$ can be written as $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$
So, $\sqrt{\dfrac{16}{36}}$ can be written as $\sqrt{\dfrac{16}{36}}=\dfrac{\sqrt{16}}{\sqrt{36}}$
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 $16=2\times 2\times 2\times 2$
The prime factors of $36=2\times 2\times 3\times 3$
Now, expressing the numerator and the denominator of our expression as their prime factors, we get
$\dfrac{\sqrt{16}}{\sqrt{36}}=\dfrac{\sqrt{2\times 2\times 2\times 2}}{\sqrt{2\times 2\times 3\times 3}}$
Since there are two pairs of ‘2’ and in the numerator and one pair of ‘2’ and one pair of ‘3’ in the denominator so taking two ‘2’s from the numerator and one ‘2’ and one ‘3’ from denominator for the square root, we get
$\dfrac{\sqrt{2\times 2\times 2\times 2}}{\sqrt{2\times 2\times 3\times 3}}=\dfrac{2\times 2}{2\times 3}$
Cancelling out ‘2’ both from the numerator and the denominator, we get
$=\dfrac{2}{3}$
Hence, $\sqrt{\dfrac{16}{36}}=\dfrac{2}{3}$
This is the required solution of the given question.

Note: Since both ‘16’ and ‘36’ are two square terms which can be expressed as the square of ‘4’ and ‘6’ respectively, i.e. $16={{\left( 4 \right)}^{2}}$ and $36={{\left( 6 \right)}^{2}}$ so the given expression can be simplified directly as $\sqrt{\dfrac{16}{36}}=\dfrac{\sqrt{{{\left( 4 \right)}^{2}}}}{\sqrt{{{\left( 6 \right)}^{2}}}}=\dfrac{4}{6}=\dfrac{2}{3}$. This is the alternative solution.