How do you find the square root of 500?

Hint:
Recall that "square root of a" means $ {{a}^{\dfrac{1}{2}}} $ , which is also written as $ \sqrt{a} $ . For positive real numbers a and b, $ \sqrt{a} $ × $ \sqrt{b} $ = $ \sqrt{ab} $ (see Note below). Express 500 as a product of prime numbers and see if it can be simplified or not. The following rule of exponent $ {{a}^{\dfrac{p}{q}}} $ = $ \sqrt[q]{{{a}^{p}}} $ is useful.

Complete Step by step Solution:
We can write 500 = 2 × 2 × 5 × 5 × 5 = $ {{2}^{2}} $ × $ {{5}^{2}} $ × 5.
Note that we only keep the even powers of the numbers so that it's easy to take their square root (separate them from the radicals).
Taking the square root, we get:
 $ \sqrt{500} $ = $ \sqrt{{{2}^{2}}\times {{5}^{2}}\times 5} $
We know that for positive real numbers a and b, $ \sqrt{a} $ × $ \sqrt{b} $ = $ \sqrt{ab} $ , therefore:
⇒ $ \sqrt{500} $ = $ \sqrt{{{2}^{2}}} $ × $ \sqrt{{{5}^{2}}} $ × $ \sqrt{5} $
Using $ {{a}^{\dfrac{p}{q}}} $ = $ \sqrt[q]{{{a}^{p}}} $ , we have:
⇒ $ \sqrt{500} $ = $ {{\left( {{2}^{2}} \right)}^{\dfrac{1}{2}}} $ × $ {{\left( {{5}^{2}} \right)}^{\dfrac{1}{2}}} $ × $ \sqrt{5} $
⇒ $ \sqrt{500} $ = $ {{2}^{\dfrac{2}{2}}} $ × $ {{5}^{\dfrac{2}{2}}} $ × $ \sqrt{5} $
⇒ $ \sqrt{500} $ = 2 × 5 × $ \sqrt{5} $
⇒ $ \sqrt{500} $ = 10 $ \sqrt{5} $ , which is the required simplification.
Using the value of $ \sqrt{5} $ = 2.236..., we find that:
⇒ $ \sqrt{500} $ = 10 × 2.236...

⇒ $ \sqrt{500} $ = 22.36..., which is the required answer.

Note:
If a and b are negative, then the rule $ \sqrt{a} $ × $ \sqrt{b} $ = $ \sqrt{ab} $ doesn't work. In that case, we have to consider complex numbers to make the radicals positive and multiply the imaginary unit separately. For example, $ \sqrt{-3} $ × $ \sqrt{-2} $ = $ \sqrt{3}\sqrt{-1} $ × $ \sqrt{2}\sqrt{-1} $ = $ -\sqrt{6} $ .
In general, the notation $ {{a}^{x}} $ is used to represent the value of the product a × a × a × a ... (x times). Here a is called the base (radix) and x is called the exponent / power (index). If $ {{a}^{x}} $ = b, then we say $ {{b}^{\dfrac{1}{x}}} $ = $ \sqrt[x]{b} $ = a, which is read as "x-th root of b is equal to a". The roots are also called radicals.
When representing the numbers, for radical numbers $ a\sqrt{b} $ means "a × $ \sqrt{b} $ ", whereas for fractions $ a\dfrac{b}{c} $ means "a + $ \dfrac{b}{c} $ ".