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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Example 30: Find the square root of:

(a) 176.89 (b) 4395.69 (c) 0.053361

Solution: (a) 13.3 (b) 66.3 (c) 0.231 [The integral part of the
decimal number is zero,
1 1 76.89 6 43 95.69 2 0.05 33 61 so, the integral part of
–1 –36 –4 square root will also be
zero.]
23 76 126 795 43 133
–69 –756 –129

263 789 1323 3969 461 461
–789 –3969 –461
0 0 0
.
\ 176 89. = 13 3 \ 4395 69. = 66 3 \ 0 053361 = 0 231
.
.
.
Square Root of Product of Two Numbers and Fractional Numbers
• If a and b are two positive numbers, the product of the square roots of
a and b is equal to the combined product of square root a and b. Remember
a + b ≠ ( a + b)
a × b = ab or ab = a × b
a − b ≠ ( a − b)

For example, 9 × 4 = 3 × 2 = 6

or, 9 × 4 = 36 = 2 × 2 ×× = 2 × 3 = 6
3
3
• If a and b are two numbers in fractional form, the square root of the fraction is equal to the
quotient of the square roots of the numerator and the denominator.

a = a or a = a
b b b b

16 16 4
For example, = =
25 25 5

Estimating Square Roots

Can we find the square root of only perfect squares? What about non-perfect squares, like 360,
525, etc.?

We can estimate the square root of such numbers by finding a number whose square is closest to
the given number. Let us try to estimate the square root of 360.
We know that 100 < 360 < 400

Also, 100 = 10 and 400 = 20

The square root of 360 lies between 10 and 20.
Now, 18 = 324 and 19 = 361
2
2
360 lies between 324 and 361. That is
2
18 < 360 < 19 or 18 < 360 < 19
2
141 Squares and Square Roots