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- Square Number
- How to find the square of Number easily
- Pythagorean triplets
- Square Root
- How to Find Square root
- Estimating Digits in the Square Root

number, then

1=1

4=2

(a+b)

23

=

25

So, 2

6,8,10

6

Find a Pythagorean triplet in which one member is 12.

If we take

Then,

Then the value of

So, we try to take

So, let us take 2

then

Thus,

Therefore, the required triplet is 12, 35, 37

So we know that

m=n

Square root of m

√m =n

Square root is denoted by expression √

Consider 36

Then,

(i) 36 – 1 = 35 (ii) 35 – 3 = 32 (iii) 32 – 5 = 27 (iv) 27 – 7 = 20

(v) 20 – 9 = 11 (vi)11 – 11 = 0

So 6 odd number, Square root is 6

We will get same prime number occurring in pair for perfect square number. Square root will be given by multiplication of prime factor occurring in pair

Consider

81

81= (3×3)×(3×3)

√81= 3×3=9

Find the square root of 57600.

Therefore 57600= 2 × 2 × 2 × 2 × 5×3 = 240

In the below example 4 < 6, So taking 2 as divisor and quotient and dividing, we get 2 as reminder

In the below example we bring 25 down with the reminder, so the number is 225

In the below example, it will be 4

In this case 45 × 5 = 225 so we choose the new digit as 5. Get the remainder.

In case of Decimal Number, we count the bar on the integer part in the same manner as we did above, but for the decimal part, we start pairing the digit from first decimal part.

**Example**
Find the square root of 870.25

**Solution**

Find the Square root of 841

if

Or

We can create the pair in the number, the Number of pair gives the number of digit in the square root

Find the number of digits in the square root of the following numbers.

(i) 25921

(ii) 37249

Here n is odd, so (n+1)/2 = 3

So three digits will be present in square root

Perfect square is a number which can be expressed as square of any integers $2 \times 2 =4$ $3 \times 3 =9$

33 is not perfect square as it can be expressed as square of an integers

256 is a perfect square as $256=16^2$

Cube root of a number can be found using prime factorization method

1000 is not perfect square as it can be expressed as square of an integers

You can the below chart to estimate the unit digit

You can below trick to find the square roor quickly
Step 1: if a perfect square is of n-digits, then its square root will have n/2 digits if n is even or (n+ 1)/2 if n is odd. So for three digit number, square root will be 2 digit, 4 digit number square root will be 2 digit number,5 digit number square root will be 3 digit number
Step 2 We can predict the Unit digit of the square root from the unit digit of the number. If the unit digit is 4, then the square root unit digit can be 2 or 8
Step 3: __For Three Digit Number__
(i)Now we take the right most number and let call it n
(ii) Now we determine two squares this number lies, $a^2$ < n < $b^2$. We choose the smallest one i.e a.
(iii) If we have two option for unit digit,then we can decide using the below method
(iv) We multiply a and b ,if ab > n, then take the lessor number else take the bigger number
Example 784
Here Unit of square root can be 2 or 8
Now $2^2 < 7> 3^2$, so we choose the smaller number
Now the square root can be 22 or 28
Now $2 \times 3 =6$ < 7, so we need to take bigger number
$\sqrt {784} =28$
Step 4: __For four Digit Number__
(i)Now we take the right most number 2 digit and let call it n
(ii) Now we determine two squares this number lies, $a^2$ < n < $b^2$. We choose the smallest one i.e a.
(iii) If we have two option for unit digit,then we can decide using the below method
(iv) We multiply a and b ,if ab > n, then take the lessor number else take the bigger number
Example 6889
Here Unit of square root can be 3 or 7
Now $8^2 < 68 > 9^2$, so we choose the smaller number 8
Now the square root can be 83 or 87
Now $8 \times 9 =72$ > 68, so we need to take less number
$\sqrt {6889} =83$
Step 4: __For five Digit Number__
(i)Now we take the right most number 3 digit and let call it n
(ii) Now we determine two squares this number lies, $a^2$ < n < $b^2$. We choose the smallest one i.e a.
(iii) If we have two option for unit digit,then we can decide using the below method
(iv) We multiply a and b ,if ab > n, then take the lessor number else take the bigger number
Example 10816
Here Unit of square root can be 4 or 6
Now $10^2 < 108 > 11^2$, so we choose the smaller number 10
Now the square root can be 104 or 106
Now $10 \times 11 =111$ > 108, so we need to take less number
$\sqrt {10816} =104$

100489

- If a natural number m can be expressed as $n^2$ , where n is also a natural number i.e $m =n^2$ then m is a square number.
- All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place.
- Square numbers can only have even number of zeros at the end.
- Square root is the inverse operation of square. There are two integral square roots of a perfect square number.Positive root is denoted by $\sqrt $

**Notes****Worksheets & Ncert Solutions**

Class 8 Maths Class 8 Science