Page 46 - ICSE Math 8

3. The square root of an even square number is always even.
4. The square root of an odd square number is always odd.
5. If a number has a natural number as its square root, then it must end with 0, 1, 4, 5, 6 or 9.

Units digit of square 0 1 4 5 6 9

Units digit of square root 0 1 or 9 2 or 8 5 4 or 6 3 or 7

6. Negative numbers do not have square roots in the system of rational numbers.

Finding Square Root by Repeated Subtraction
Step 1: Do repetitive subtraction of 1, 3, 5, 7, 9, ... from the given number until the result is 0.
Step 2: Count the number of times subtraction has been performed to arrive at 0. Let this number be n.
Step 3: Square root of the number = n
Example 11: Find the square root of 49 by repeated subtraction.
Solution: 1. 49 – 1 = 48 2. 48 – 3 = 45 3. 45 – 5 = 40 4. 40 – 7 = 33
5. 33 – 9 = 24 6. 24 – 11 = 13 7. 13 – 13 = 0

Repeated subtraction is done 7 times. \ 49 = 7

Finding Square Root by Prime Factorization Method
2 9,216
Step 1: Break the given number into its prime factors by repetitive division. 2 4,608
Step 2: Make pairs of prime factors till all factors are exhausted. 2 2,304
Step 3: Take one factor from each pair. 2 1,152
Step 4: Multiply all the factors taken in step 3. 2 576
Step 5: The resultant product is the square root of the given number. 2 288

Example 12: Find the square root of 9,216 by prime factorization method. 2 144
Solution: Writing 9,216 as a product of its prime factors, we get 2 72
9,216 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 2 36
\ 9,216 = 22 22 22 22 22 33× × × × × × × × × × × 2 18
3 9
= 2 × 2 × 2 × 2 × 2 × 3 3 3
= 96 1
\ 9,216 = 96 ( Taking one factor from each pair and
finding the product )
Example 13: 2,025 plants are to be planted in a garden in such a way that each row contains as many plants
as the rows. Find the number of rows and the number of plants in each row?
Solution: Let the number of rows = x 5 2,025
\ Number of plants in each row = x 5 405
\ Total number of plants = x × x = x 2
2
According to the question, x = 2,025 3 81
3 27
3
3
fi x = 2 025, = 5 ¥ 5 ¥¥ 3 ¥¥ 3 3 9
= 5 × 3 × 3 = 45 3 3
So, the number of rows = 45 and the number of plants in each row = 45. 1

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