Question

How do you simplify $\sqrt {\dfrac{{75}}{4}} $ ?

Answer 1

Hint: Here we can simplify this numerical root by finding the perfect squares of each term and then eliminating the square root. And to simplify the term, we get the required answer.

Complete step-by-step solution:
When we are need to simplify numerical root, we should look for these two key facts:
$\sqrt {{a^2}} = a$
$\sqrt {ab} = \sqrt a \sqrt b $
So, anytime we have a number inside a root , we should try to write it as a product of other numbers, of which at least one is a perfect square. Let’s analyze your case.
First of all, using the second property, we can write,
$ \Rightarrow \sqrt {\dfrac{{75}}{4}} = \dfrac{{\sqrt {75} }}{{\sqrt 4 }}$
In fact, every fraction can be read as multiplication using, $\dfrac{a}{b} = a.\dfrac{1}{b}$
Now let’s deal with the two roots separately. We would start with the denominator, since we already have a perfect square under a square root, so they simplify we have,
$ \Rightarrow \dfrac{{\sqrt {75} }}{{\sqrt 4 }} = \dfrac{{\sqrt {75} }}{2}$
As for$\sqrt {75} $, we can see that $75 = 25 \times 3$ and $25 = {5^2}$ is a perfect square. So by the second rule above, we have $\sqrt {75} = \sqrt {25 \times 3} = \sqrt {25} \times \sqrt 3 = 5\sqrt 3 $
This leads to the final answer,
$ \Rightarrow \sqrt {\dfrac{{75}}{4}} = \dfrac{{5\sqrt 3 }}{2}$
But how do we find the most appropriate way to rewrite our number, in this case $75 = 25 \times 3$, we can use the prime factorization.
$75 = 3 \times {5^2}$ and select only the primes with even exponent. In this case ${5^2}$ .

Hence, $\dfrac{{5\sqrt 3 }}{2}$ is the required answer.

Note: Generally, you don’t want to have radicals at the denominators. So let’s say that we want to simplify the expression $\dfrac{{\sqrt a }}{{\sqrt b }}$ where $a$ and $b$ can be any expression you want. Since, of course $\dfrac{{\sqrt b }}{{\sqrt b }} = 1$.
We can multiply it without changing the value of our expression, so we have $\dfrac{{\sqrt a }}{{\sqrt b }} = \dfrac{{\sqrt a }}{{\sqrt b }}.\dfrac{{\sqrt b }}{{\sqrt b }}$. The advantage is that now we observe that $\sqrt b .\sqrt b = b$ and so our expression becomes $\dfrac{{\sqrt {ab} }}{b}$ , and we got rid of the radical at the denominator.