Question

What is the square root of $80$ over$125$ ?

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 then simplify the term.

Complete step-by-step solution:
When we are need to simplify numerical root, we must 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. Now for our case,
First of all, we can write the given question like this,
\[ \Rightarrow \sqrt {\dfrac{{80}}{{125}}} \]
Using second property, we have,
$\sqrt {\dfrac{{80}}{{125}}} = \dfrac{{\sqrt {80} }}{{\sqrt {125} }}$
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 numerator,
As for$\sqrt {80} $ , we can see that $80 = 16 \times 5$ and $16 = {4^2}$ is a perfect square. So by the second rule above, we have $\sqrt {80} = \sqrt {16 \times 5} = \sqrt {16} .\sqrt 5 = 4\sqrt 5 $
Similarly for the denominator, that is $\sqrt {125} $ , we can see that $125 = 25 \times 5$ and $25 = {5^2}$ is a perfect square. So by the second rule above, we have $\sqrt {125} = \sqrt {25 \times 5} = \sqrt {25} .\sqrt 5 = 5\sqrt 5 $
Which leads to the final answer,
$\sqrt {\dfrac{{80}}{{125}}} = \dfrac{{4\sqrt 5 }}{{5\sqrt 5 }}$
On dividing, the above we get,
 $\sqrt {\dfrac{{80}}{{125}}} = \dfrac{4}{5}$

Note: Generally, we 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 that we have. 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.