Page 62 - Start Up Mathematics_8 (Non CCE)

Square Roots

Square root of a number n is that number which when multiplied by itself gives n as the product. In other
2
words, if n is the given number and m is its square root, then n = m . Square root is represented by the symbol
, i.e., n = . m
2
2
2
4
For example, 4 = 16 fi 16 = 5 = 25 fi 25 = 13 = 169 fi 169 = 13
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Properties of square roots
Let’s continue with what you have learnt about the properties of squares and apply them to the properties of
square roots.
I. A number ending with 2, 3, 7 or 8 does not have a natural number as its square root.
II. A number ending with odd number of zeros does not have a natural number as its square root.
However, if a number ends with even number of zeros, its square root is a natural number ending with
half the number of zeros than the given number.
III. The square root of an even number is always even.
IV. The square root of an odd number is always odd.
V. 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
VI. Negative numbers do not have square roots in the system of rational numbers.

Methods to find square roots
1. 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
2. Finding Square Root by Prime Factorization Method 2 9,216
Step 1: Break the given number into its prime factors by repetitive division.
Step 2: Make pairs of prime factors till all factors are exhausted. 2 4,608
Step 3: Take one factor from each pair. 2 2,304
Step 4: Find the product of all the factors taken in step 3. 2 1,152
Step 5: The resultant product is the square root of the given number. 2 576
Example 12: Find the square root of 9,216 by prime factorization method. 2 288

Solution: Writing 9,216 as a product of its prime factors, we get (NCERT) 2 144
9,216 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 2 72

\ 9,216 = 2 2 2 2 2 2 2 2 2 2 3 3¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 2 36
2 18
= 2 × 2 × 2 × 2 × 2 × 3 3 9
= 96 3 3

\ 9,216 = 96 ( Taking one factor from each pair and 1
finding the product )
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