Approximating Square Roots

The ‘square root’ of a given positive number ‘n’ is that number which, when multiplied by itself, gives ‘n’.

If the square root of a given number is not a whole number, then we can approximate the square root of a number.

Square Roots of Perfect Squares

$\\sqrt{1}$	1
$\\sqrt{4}$	2
$\\sqrt{9}$	3
$\\sqrt{16}$	4
$\\sqrt{25}$	5
$\\sqrt{36}$	6
$\\sqrt{49}$	7
$\\sqrt{64}$	8
$\\sqrt{81}$	9
$\\sqrt{100}$	10
$\\sqrt{121}$	11
$\\sqrt{144}$	12

Steps to approximate square roots:

Step 1: Think of two perfect squares between which the given number lies.

Example: If we want to find the square root of 12, then 3² = 9 and 4² = 16. So, $\\sqrt{12}$ lies between 9 and 16.

Step 2: Divide the given number by one of the two square roots in step 1.

12 ÷ 3 = 4

Step 3: Find the average of the root and the result in step 2.

$\\frac{4 + 3}{{2}}$ = $\\frac{7}{{2}}$ =3.5

Step 4: Repeat steps 2 and 3 until you reach the desired result.

Repetition of step 2: 12 ÷ 3.5 = 3.4285

Repetition of step 3: = $\\frac{3.4285+3.5}{{2}}$ = $\\frac{6.9285}{{2}}$ = 3.46425

Now you need to decide the level accuracy to find the square root of given number. If you want it to be more accurate, keep on repeating steps 2 and 3. We leave $\\sqrt{12}$ here.

CHECK POINT

Find the square root of the given numbers, rounded to the nearest tenth.

$\\sqrt{165}$
$\\sqrt{189}$
$\\sqrt{44}$
$\\sqrt{103}$
$\\sqrt{114}$

Answer key

12.8
13.7
6.6
10.1
10.7

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