Question

How do you simplify $\sqrt {500} $?

Answer 1

Hint: We must know that in order to solve such problems where we need to simplify the term that is under the square root, we need to know what actually square root means and the way by which we can take out the terms that are there in the root and have the perfect square.

Complete step-by-step answer:
Here we are given to find the square root of the number that is $500$ and therefore we need to know what actually square root means. So the square root of any number represents the number that has been multiplied by itself in order to get the number that is contained in the root.
This can be made clearer with an example. For example: If we have the number $4$ and we need to find its square root then we need to find the number that will give the result $4$ when that number is multiplied by itself. So we know that when $2$ is multiplied by itself we get $(2)(2) = 4$ so we can write this as $\sqrt 4 = \sqrt {(2) \times (2)} $ and whenever we have a pair of any number in the root we can take it outside the root so we will get $\sqrt 4 = \sqrt {(2) \times (2)} = 2$
Now similarly we can factorise $500$ and get the value of $\sqrt {500} $
Now we know
$
  500 = 2 \times 250 \\
  \Rightarrow 250 = 2 \times 125 \\
  \Rightarrow 125 = 5 \times 25 \\
  \Rightarrow 25 = 5 \times 5 \\
  \Rightarrow 5 = 5 \times 1 \\
 $
So we can write $500$ in the form:
$500 = 2 \times 2 \times 5 \times 5 \times 5$
Putting this value in the given problem, we will get:
$\sqrt {500} = \sqrt {2 \times 2 \times 5 \times 5 \times 5} $
Now as we know that we have the two pairs in the square root of the number $500$
So we can write the number that is in the pair once outside the square root.
So we have \[2{\text{ and 5}}\] in the root that are in pairs. So we can say take them outside the root we will get:
$\sqrt {500} = \sqrt {2 \times 2 \times 5 \times 5 \times 5} = (2)(5)\sqrt 5 $
So we will get $\sqrt {500} = 10\sqrt 5 $
Hence the simplified form of $\sqrt {500} = 10\sqrt 5 $

Note: Here in these types of problems the students must keep in mind that whenever we are given the root of any number that we need to simplify, we need to factorise the number inside it and then write the terms that are in pair outside the bracket once and the remaining inside the root.