Square & Square Roots Square : If a number is multiplied by itself then the product is the square of the number. Thus the square of 5 is 5x5 = 25
2.
219961 = 469 Find the square root of 59.1361
2
7.69
1 1 1 1 eg. x 2 2 2 4
7
59 . 13 61 49
2
2 2 4 2 x 3 3 3 9
146
1013 876
Square root: The square root of a number is one of two equal factors which is multiplied together gives that number.
1529
1 37 6 1 0
eg. 121 = 11x11 = 11
59.1361 = 7.69
10000 = 100 x 100 = 100
Properties of a perfect square
Finding Square root by means of factorisation When a given number is a perfect square, we resolve it into prime factors and take the product of prime factors, choosing one out of each pair. eg. Find the square root of 1156
No perfect square ends with 2,3,7,8
No perfect square ends with an odd number of zeros.
The perfect square consisting of (n-1) or n digits will have n 2 digits in their root
Factors of 1156 is 2x2x17x17 1 1 5 6 = 2 x 2 x 1 7 x 1 7 = 2 2x 1 7 2 1156 = 2 2 x 17 2 = 2x17 = 34 General method to find the square root In the given number mark off the digits in pairs, from right and then find the square root as shown in the example below. eg. 1. 4 86 929
Find the square root of 219961 469 21, 99, 61 16 599 516 8361 8361 0
1 37 6 1
The square of a number other than unity is either a multiple of 4 or exceeds a multiple of 4 by 1.
CUBE ROOT The cube root of a number is one of three equal factors which if multiplied gives that number. Cube root of a number can be found out by using the following steps. 1 . Write down all the prime factors of the given numbers. 2 . Write the prime factors in the index notation, ie, in a n form. 3 . Divide the index by 3, then the result will be the cube root of the given number.
eg. 1. Find the cube root of 512 512=(2x2x2) x (2x2x2) x (2x2x2)
3.
9 13 3 512 = (2 ) 2 8 2. Find the cube root of 0.000027
3
3.
3
27 0.000027 1000000
13
33
13
3 13
100
3 0.03 100 Learn by heart the following square roots
=
4.
If
x 4 , the value of x is 49 7
a)
7
If x =
Square
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Cube
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225
1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375
No 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
a)
2= 1. 41421
6 =2. 44949
3= 1. 73205
7 = 2. 64575
5=2. 2360
10 = 3.1622
1 . If x = 3018 + 36+169, the value of x is
2.
44
If x = a)
3
b) 5 5
c) 6 9
d) 4 3
169 . x 0.9 , the value of x is 13 . x0.13
b) 1 3
c) 3 9
a)
6.
If
9.
b) 1 0
5 8
c) 8
d) 1 5
324 x 0.4 ? 9 b)
8 5
c)
9 5
d)
10 5
169 169 . , the value of x is x
a)
100
b) 1 0 0 0
c)
10000
d) 1, 00, 000
144 324 104 x ? 6 6 169
7.
8.
12
a)
26
b) 1 4
c)
10 1. 82
d) 3 6
8
? + 44 = 25% of 400
a)
3
If x = a)
b) 3 6
c) 4 9
d) 1 6
81 , the value of x is 0.09
3
b) 3 0
c) 3 0 0
d) 0 . 3
1 0 . The largest of four digit numbers which is a perfect squre is
PRACTICE TEST
a)
d) 4
196 18 65 the value 7 324 169
484 x 11
5.
Square 256 289 324 361 400 441 484 529 576 625 676 729 784 841 900
c) 1 6
of x is
SQUARES AND CUBES No.
b) 4 9
d) 0 . 3 9
a)
9801
b) 9 9 0 4
c)
9804
d) 9 8 0 9
1 1 . A certain number of people collected Rs. 125. If each person contributed as many five paise as they are in number, the number of person were a)
25
b) 5 0
c) 1 0 0
d) 1 2 5
1 2 . A gardener plants an orchard with 5776 trees. In each row there were as many trees as the number of rows. Find the number of rows. a) 7 6 b) 9 6 c) 6 6 d) 1 8 6 1 3 . Each student in a class contributed as many rupees as the number of studnets in the class for a picnic. The school contributed Rs. 150 per teacher who led the trip. If the total amount collected was Rs. 1350 and the number of teachers who led the trip was 3, how many students were there in that class? a)
36
b) 3 5
c) 3 4
9 55
b)
7 5 a) c)
1 8 . If then
16x10 - 5
d) 16x10 - 6 1
1
1
(a) -1
b) 0
c) 1
d) 3 2 3
1
2 0 . The value of
2 1 . if value of a) 2 2 . If
41
1
1 6 8 1 is
400
b) 2 1
c) 3 1
5 2.2 4 and
d) 5 1
6 2 .4 5 , the
2 5 is 3 6 1 . 3 7 b) 1 . 5 7
c) 1 . 7 3 d) 1 . 7 5
1 5 6 2 5 1 2 5 then the value of
1 5 6 2 5 1 5 6 .2 5 15 . 6 2 5 is
81 ? 121
11 45
c) 4
a) c)
1. 38 75 13 8. 75
2 3 . If
1 45 d) 11 11
b) 13 . 8 75 d) 15 6. 25
4 0 9 6 = 64 to then the value of
40.96 0.4096 0.004096 0.00004096
7 5 17.
c)
a)
x 13 then x is equal to. 144 12 0 b) 1 2 c) 1 3 d) 2 5
a)
b) 16x10 - 3
1 5 . If 1
169 4 1 6 . 225 9
16x10 - 4
1 9 . The value of 2 1 3 2 2 3 is
d) 3 0
1 4 . Some persons contributed Rs. 1089. Each person gave as many rupees as they are in number. Find their number. a) 3 3 b) 6 6 c) 4 5 d) 2 3
a)
a)
is
is equal to
6 35 2
a) 7 . 0 9 c) 7. 11 04 2 4 . The expression
b) 6 35 d) 1
0.04 x 0.4 xa = 0.004 x 0.4 x
2
b,
a is b
2
b) 7. 10 14 d) 7 . 1 2
1 2 2
1 equals 2 2
a)
2
b)
c)
2 2
d) 2 2
2 2
ANSWERS TO PRACTICE TEST 1.
(b)
2.
(a)
9.
(b)
1 0 . (a)
1 7 . (a)
1 8 . (c)
3.
(c)
4.
(c)
5 . (b)
6.
(c)
7.
(a)
8.
(c)
1 1 . (b)
1 2 . (a)
13. (d)
1 4 . (a)
1 5 . (d)
16.(b)
1 9 . (c)
2 0 . (b)
21.(c)
2 2 . (c)
2 3 . (c)
2 4 . (a)
Laws of Exponents A long product axaxa ....m factors can be expressed, in short by notation a m, where `a' is called the base and `m' the power (or index or exponent) Thus axaxa.... 10 factors = a 10.
iv).
abm
am . bm
v). a 1
Definition am: If a is a natural number, then am stands for the product of `m' factors each equal to `a'.
m vi). a
1 am
Property 1: a m x a n = a m+n
SOLVED EXAMPLES :
(eg. a 3xa 4 = a 3+4 = a 7 )
1.
81 3 x 161 4 x 2 2 ?
m-n
Property 2 : a n=a (eg. = a 5-3 = a 2 ) a3 am Property 3 : (am )n amn a5
3 13
2.
Meaning of a° : Any non-zero number raised to zero power is equal to 1.
m
1 m a
1 ) a3
3
2
3
3.
2 2
3 x 3 2
4.
Fractional Index : a
4
2 7
Evaluate
a 3 3 2 (2 7 )2
eg. a1 2
i).
am xan am n
ii).
am am n an
iii).
a
m n
3
3 a , a2 3 a2
If `a' and `b' are non-zero integers, `m' and `n' are rational numbers, then
amn
= 1
5.
4
34 x 3 31 2 1 312 31 2
3 4
n
26 ?
8
34 x 38
3 1n
2
2 x 2
26 x 26 26 64 26
ie. a° = 1 where a 0 eg. 5° = 1.
(eg. a3
x 2 2
2 1 x 2 1 x 2 2 2 112 2 1
(eg. (a2 )3 a6
Negative Index : If a 0 then a
4 14
2 x 2
1 32
3
3
2 3
9 3 26 25 13
2
6
1 2
3 2 5
1 2
=
1
1 5 32 2 2 9 3
=
1
1 5 7 7 9 4 14 1 15 9 3 9 9
If 5 a 3 1 2 5 , find the value of 5 a 3 5a 3125 5a 55
9.
a 5 6.
5 a3 5 5 3 5 2 2 5 If m and n are whole numbers such that
mn 121 , find the value of (m 1)n1 mn 121 112 m 11, n 2 n 1
2 1
(m 1) 1. 2.
3.
(6 4 )
3
b) 2 2
2
a)
1 16
If
n
(3 2 )
4
c) 1 5
d) 2
equals
1 8
b)
c)
3 16
7.
8.
x 3 4 1
a)
b) 3
1
2 x 1 If 2
5
b) 2 5
c) 5
0
x 1
1
x 2 2 5
1
b) 4
c)
3
1 2 n
x9 35n
b a
b)
is
d)
1 32
1 3 . If 2 x 1 2 x 1 1280 ,then the value of x is a)
d) 2
7
b) 8
c) 9
a)
1 25
b)
c)
25
d) -25
d) 1 0 2 3
gives
1 25
2 n 7
1 5 . The units digit in 2 4 6 9 1 5 3 will be
is
1 c) 9
a)
2 d) 3
16.
x 3
1 2
1 8
1 1 4 . The simplification of 1 25
d) 7
c) -2
9 b) 13
If b
721
c) 5
b) -1
1 3 a
72
d) 5 5
1 2 . (4 )0 .5 x (0 .5 ) 4 is equal to
d) 2
1 , the value of x is 8 x 3
The value of
a)
(125 )0.25 is equal to
2 64 , the value of n is
2 4 6 7
a)
4x
1 1 . If a3 b ab3 c 180 and a, b, and c are positive integers, the value of c is a) 1 0 b) 1 5 c) 2 5 d) data inadequate
a)
153
a)
1
a)
a) 2 b) 4 c) 6 d) 1 2 4. If 1.125x10 k=0.001125,the value of k is a) -4 b) -3 c) -2 d) -1 5 . The unit's digit in the product of
6.
between the obtained and the actual value is a) 2 1 x 9 2 b) 2 2 x 9 3 c) 2 3 x 9 4 d) Zero
(1 1 1) 1 0 1 0 0 0 0 PRACTICE TEST
23 1
2 x 9 2 . He wrote it as 2592. The difference
10. 5
25 25 is divided by 24, the remainder is a)
A boy was asked to write the value of
5
b) 3
c) 7
2 4 3 0 .1 2 x 2 4 3 0 .0 8 a)
, x is equal to
1
3
b) 9
d)
9
d)
27
equals c) 1 2
1 7 . The value of 4 7 1 6 4 x 1 6 is euqal to: c)
7 2
d)
2
a)
1 16
b)
1 4
c) 4
d)
1
ANSWERS TO PRACTICE TEST 1.
(c)
2.
9.
(d)
1 0 . (c)
1 7 . (d)
(a)
3.
(d)
1 1 . (d)
4.
(b)
5 . (c)
6.
(d)
1 2 . (c)
13. (c)
1 4 . (c)
7.
(c)
1 5 . (d)
8.
(d)
1 6 . (a)
