Question

How do you simplify $\sqrt {40} $?

Answer 1

Hint: Here we must know what square root actually means. Square root of any number means we need to find the number which when multiplied by itself gives us the number inside the root. Similarly here we need to find the number which when multiplied by itself gives us $40$ .

Complete step-by-step answer:
Here we are given to find the square root of $40$
We must know that when we are asked to find the square root of any number we actually need to find the number which when multiplied by itself gives us that number present inside the root. Here we need to find the number which gives us $40$ when multiplied by itself. The example will make it clearer. If we need to find the square root of the number $9$ so need to find the number which when multiplied by itself gives us the number $9$
We know that $3 \times 3 = 9$ so its answer is $3$ as when $3$ is multiplied by itself the result is $9$
Similarly we need to find the square root of the term $40$ which can be written as $\sqrt {40} $
So we need to know that:
$
  40 = (2)(20) \\
  20 = (2)(10) \\
  10 = (2)(5) \\
  5 = (5)(1) \\
 $
From the above factors of $40$ we get to know that we can write $40$ in the form of its factors as:
$40 = (2)(2)(2)(5)$
So we need to find the value of $\sqrt {40} = \sqrt {(2)(2)(2)(5)} $
In such problems where we have terms inside the root, we must know that if the same term inside the root is occurring twice, we can write it once outside the root.
As here we can see that one pair can be made of the numbers $2$ which is inside the root and one $(2)(5) = 10$ will be left inside. So we can take the number that is in pair outside the root and we will get:
$\sqrt {40} = \sqrt {(2)(2)(2)(5)} = (2)\sqrt {2 \times 5} = 2\sqrt {10} $
We also know that approximate value of $\sqrt {10} = 3.16$
So we can simplify it further as:
$2(3.16) = 6.32$
So we get that $\sqrt {40} = 6.32$.

Note: Here if we would have been given to find the square of $2\sqrt {10} $ then we would have multiplied it twice and then we would have written the answer which would be $40$ and therefore the student must know the difference between square of the number and square root of the number.