IMPORTANT FACTS AND FORMULAE
Square
Root: If x2 = y, we say that the square root of y is
x and we write, √y = x.
Thus, √4 = 2, √9 = 3, √196 = 14.
Cube Root:
The cube root of a given number x is the number whose cube is x. We denote the
cube root of x by 3√x.
Thus, 3√8 = 3√2 x 2 x 2 = 2, 3√343
= 3√7 x 7 x 7 = 7 etc.
Note:
1.√xy = √x * √y 2. √(x/y) = √x / √y = (√x / √y) * (√y / √y) = √xy / y
SOLVED EXAMPLES
Ex. 1. Evaluate √6084 by
factorization method .
Sol. Method: Express the given number as
the product of prime factors.
2 6084
Now, take the product of these
prime factors choosing one out of 2 3042
every pair of the same primes. This
product gives the square root
3 1521
of the given number.
3 507
Thus, resolving 6084 into prime
factors, we get: 13 1696084 = 22 x 32
x 132 13 \ √6084
= (2 x 3 x 13) = 78. 
Ex.
2. Find the square root of 1471369.
Sol. Explanation:
In the given number, mark off the digits 1
1471369 (1213
in pairs starting from the unit's digit. Each pair and 1
the remaining one digit is called a period. 22 47
Now, 12 = 1. On subtracting, we get 0 as remainder. 44
Now, bring down the next period i.e., 47. 241 313
Now, trial divisor is 1 x 2 = 2 and trial dividend is 47. 241
So, we take 22 as divisor and put 2 as quotient. 2423 7269
The remainder is 3.
7269
Next, we bring down the next period which is 13. x
Now, trial divisor is 12 x 2 = 24 and trial dividend is
313. So, we take 241 as dividend and 1 as quotient.
The remainder is 72.
Bring down the next period i.e., 69.
Now, the trial divisor is 121 x 2 = 242 and the trial
dividend is 7269. So, we take 3as quotient and 2423
as divisor. The remainder is then zero.
Hence,
√1471369 = 1213.
Ex. 3. Evaluate: √248 + √51 + √ 169 . 
Sol. Given expression = √248 + √51 + 13 = √248
+ √64 = √ 248 + 8 = √256 =
16.Ex. 4. If a * b * c = √(a + 2)(b + 3)
/ (c + 1), find the value of 6 * 15 * 3.
Sol. 6 * 15 * 3 = √(6 + 2)(15 + 3) / (3 + 1) = √8 * 18 / 4 = √144 /
4 = 12 / 4 = 3. Ex. 5. Find the value of √25/16.
Sol. √ 25 / 16 = √ 25 / √ 16 = 5 / 4
Ex. 6. What
is the square root of 0.0009?
Sol. √0.0009= √ 9 / 1000 = 3 / 100 = 0.03.
Ex.
7. Evaluate √175.2976.
Sol.
Method: We make even number of
decimal places 1 175.2976 (13.24
by affixing a zero, if necessary. Now, we mark off 1
periods and extract the square root as shown. 23 75
69
\√175.2976
= 13.24
262 629
524
2644 10576
10576
x
Ex.
8. What will come in place of question mark in each of the
following questions?
(i) √32.4 / ? = 2 (ii)
√86.49 + √ 5 + ( ? )2 = 12.3. Sol. (i) Let √32.4 / x = 2. Then,
32.4/x = 4 <=> 4x = 32.4 <=> x = 8.1.(ii) Let √86.49 + √5 + x2
= 12.3.
Then, 9.3 + √5+x2 = 12.3 <=> √5+x2 = 12.3 - 9.3 = 3
<=> 5 + x2 = 9 <=> x2 = 9 - 5= 4 <=>
x = √4 = 2.
Ex.9. Find the value of √ 0.289
/ 0.00121. |
 |
 |
Sol. √0.289 / 0.00121 = √0.28900/0.00121 = √28900/121
= 170 / 11.
 |
Ex.10. If √1 + (x / 144) =
13 / 12, the find the value of x.
Sol. √1 + (x / 144) = 13 / 12 Þ ( 1 + (x / 144)) = (13 /
12 )2 = 169 / 144
Þx / 144 = (169 / 144) - 1
Þx / 144 = 25/144 Þ x =
25.
Ex. 11.
Find the value of √3 up to three places of decimal.
Sol.
1
3.000000 (1.732
1
27 200
189
343 1100
1029
3462 7100
6924 \√3 = 1.732.
Ex. 12. If
√3 = 1.732, find the value of √192 - 1 √48 - √75 correct
to 3 places
2
of decimal. (S.S.C.
2004)
Sol.
√192 - (1 / 2)√48 - √75
= √64 * 3 - (1/2) √ 16 * 3 - √ 25 * 3
=8√3
- (1/2) * 4√3 - 5√3
=3√3
- 2√3 = √3 = 1.732
 |
Ex. 13. Evaluate: √(9.5 * 0.0085 * 18.9) /
(0.0017 * 1.9 * 0.021)
Sol. Given exp. = √(9.5 * 0.0085 * 18.9) / (0.0017 * 1.9 * 0.021)
Now, since the sum of
decimal places in the numerator and denominator under the radical sign is the same, we remove
the decimal.
\ Given exp = √(95 * 85 * 18900) / (17 *
19 * 21) = √ 5 * 5 * 900 = 5 * 30 = 150.Ex. 14. Simplify: √ [(
12.1 )2 - (8.1)2] / [(0.25)2 +
(0.25)(19.95)]
Sol. Given exp. = √ [(12.1 + 8.1)(12.1 - 8.1)] / [(0.25)(0.25 +
19.95)]
=√
(20.2 * 4) /( 0.25 * 20.2) = √ 4 / 0.25
= √400 / 25 = √16 = 4.
Ex. 15. If
x = 1 + √2 and y = 1 - √2, find the value of (x2 +
y2).
Sol. x2 + y2 =
(1 + √2)2 + (1 - √2)2 = 2[(1)2 + (√2)2]
= 2 * 3 = 6.
Ex. 16. Evaluate:
√0.9 up to 3 places of decimal.
Sol.
9 0.900000(0.948
81
184 900
736
1888 16400
15104 \√0.9
= 0.948
Ex.17.
If √15 = 3.88, find the value of √ (5/3).
Sol.
√
(5/3) = √(5 * 3) / (3 * 3) = √15 / 3 =
3.88 / 3 = 1.2933…. = 1.293.
√
(5/3) = √(5 * 3) / (3 * 3) = √15 / 3 =
3.88 / 3 = 1.2933…. = 1.293.
Ex. 18. Find
the least square number which is exactly divisible by 10,12,15 and 18.
Sol. L.C.M. of 10, 12, 15, 18 = 180. Now, 180 =
2 * 2 * 3 * 3 *5 = 22 * 32 * 5.
To make it a perfect square, it must
be multiplied by 5.
\ Required number = (22 * 32
* 52) = 900.
Ex. 19. Find the greatest number of five digits which is a
perfect square.
(R.R.B. 1998)
Sol. Greatest number of 5 digits is 99999.
3
99999(316
9
61 99
61
626 3899
3756
143
\
Required number == (99999 - 143) = 99856.
Ex.
20. Find the smallest number that must be added to 1780 to make it a perfect
square.
Sol.
4 1780 (42
16
82 180
164
16 |
\ Number to be added = (43)2 - 1780 = 1849 -
1780 = 69.
Ex. 21. √2 =
1.4142, find the value of √2 / (2 + √2).
Sol. √2 / (2 + √2) = √2 / (2 + √2) * (2 - √2) /
(2 - √2) = (2√2 – 2) / (4 – 2)
= 2(√2 – 1) / 2 = √2 – 1 = 0.4142.
22. If x
= (√5 + √3) / (√5 - √3) and y = (√5 - √3) / (√5 + √3), find the value of (x2 + y2).
Sol.
x
= [(√5 + √3) / (√5 - √3)] * [(√5 + √3) / (√5 + √3)] = (√5 + √3)2
/ (5 - 3)
=(5 + 3 + 2√15) / 2 = 4 + √15.
y = [(√5 - √3) / (√5 + √3)] *
[(√5 - √3) / (√5 - √3)] = (√5 - √3)2 / (5 - 3)
=(5 + 3 - 2√15) / 2 = 4 - √15.
\ x2 + y2 = (4 + √15)2
+ (4 - √15)2 = 2[(4)2 + (√15)2] = 2 * 31
= 62.
Ex. 23. Find the cube root of 2744.
Sol. Method: Resolve the given number as the
product 2 2744 of prime factors and take the product
of prime 2 1372
factors,
choosing one out of three of the same 2 686
prime
factors. Resolving 2744 as the product of 7 343 prime factors, we get:
7 49
7
2744 = 23 x 73.
\
3√2744= 2 x 7 = 14.
Ex. 24. By what least number 4320 be multiplied to obtain
a number which is a perfect cube?
Sol. Clearly, 4320 =
23 * 33 * 22 * 5.
To make it a perfect
cube, it must be multiplied by 2 * 52 i.e,50.
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