08
4
3
3
4
8
1
2 0 Square and Square Roots
5Unit
9 712
1
Statement A
If b > 0 and a2 = b, then a =
Example
42 = 16
Example
36 = 6
b . ‘ a’ is known as the principal square root of b.
⇒
4 =
16
or
16 = 4
(not ± 6)
81 = 9
Statement B
If a3 = b, then a =
3
b . The sign convention of ‘a’ depends on whether
b > 0 or b < 0.
Example
Example
a3 = b,
⇒
a =
a3 = –b
⇒
a =–
3
27 = 3
3
− 8 = -2
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3
b
3
or
3
b = a
b or –
3
b = a.
Unit 2
Square and Square Roots
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Example 1
Example 2
Find the value of
196 .
Find the value of
Solution:
196
196
∴
=
2×2×7×7
=
=
2×2×2×3×3×3
2 2 × 72
=
2 3 × 33
=
(2 × 7)2
=
( 2 × 3)3
=
(14)2
216
=
63
14
∴
196 =
216
3
216 =
6
Find the cube root of – 1728.
Solution:
1728
1728
=
2 × 2× 2× 2× 2× 2× 3× 3× 3
=
(2 × 2 × 3)3
=
123
∴ 3 − 1728
=
– 12
Exercise 2.1
2.
8
216 .
Solution :
Example 3
1.
3
Find the value of each of the following.
(a)
152
(b)
122
(d)
(106)2
(e)
(-45)2
(c)
252
Find the square root of each of the following.
(a)
121
(b)
256
(d)
324
(e)
289
Unit 2
Square and Square Roots
(c)
225
3.
4.
5.
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Find the value of each of the following.
(a)
113
(b)
(-20)3
(d)
(21)3
(e)
3 3 × 42
(c)
(-10)2
(c)
-27
(c)
3
Find the cube root of the following.
(a)
125
(b)
729
(d)
-8
(e)
- 343
Find the value of each of the following.
(a)
3
− 125
(b)
3
512
(d)
3
1000
(e)
3
− 1000
64
2.1 Binary Number (Base Two)& Decimal Number (Base Ten)
Example - Convert 26ten into a binary number.
(i)
26
= 16 + 8 + 2
(ii)
2
26
S
E
F
T
U
2
13
r
0
1
1
0
1
0
2
6
r
1
2
3
r
0
2
1
r
1
0
r
1
∴ 26ten = 11010two
∴
r =remainder
26 ten = 11010two
Unit 2
Square and Square Roots
9
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Example Convert the binary number 10101 into a decimal number.
1 0 1 0 1
∴
S
E
F
T
U
=
1
0
1
0
1
=
1 × 16 + 0 × 8 + 1 × 4 + 0 × 2 + 1 × 1
=
16 + 4 + 1
10101 =
21 ten
Exercise 2.2
1.
2.
Convert each of the following binary number to decimal number.
(a)
111
(b)
1011
(c)
1001
(d)
11001
(e)
1101
Convert the following decimal number into the binary form.
(a)
9
(b)
15
(c)
24
(d)
37
(e)
71
2.2 Addition
When we consider addition in the binary scale we need to remember only
the following facts:
10
0
+
0
=
0
0
+
1
=
1
1
+
0
=
1
1
+
1
=
10two
Unit 2
Square and Square Roots
or
+
0
1
0
0
1
1
1
10
Example 1, (No carrying)
101
10
111
Example 2, (One carried digit)
101
+1001
1110
(Read as ‘one and one, one zero; zero, carry one, etc.’)
Example 3, (Two separate carried digits)
101
+ 101
1010
Example 4, (Three adjacent carried digits)
101
+ 11
1000
Further examples (with decimal check) :
Example 5,
10001
+ 1011
11100two
check
17
+11
28ten
Unit 2
Square and Square Roots
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Exercise 2.3
Add the following pairs of binary numbers:
1.
1010
2.
100
4.
1010
10101
5.
10010
8.
10111
1010
6.
10101
1101
110
9.
1001
11.
111
13.
1011
10001
10
101
10.
3.
110
10
7.
1001
1011
10101
12.
1
14.
1010
101
10001
1001
10011
15.
1011
10011
1110
Add the following numbers and in each case check your work by converting
both the given numbers and your answer into the decimal scale:
16.
10111
17.
1001
19.
11011
18.
1101
20.
1011
12
10101
Unit 2
Square and Square Roots
11111
101
10001
1011
21.
10011
111111
2.3 Subtraction
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When we consider subtraction in the binary scale we need to remember
only the following facts:
0
–
0
=
0
1
–
0
=
1
1
–
1
=
0
10two –
1
=
1
Examples
(i) 11
(ii) 10
(iii) 101
(iv) 101
(v) 100
–1
–1
–10
–11
–11
10
1
11
10
1
Exercise 2.4
Carry out the following subtractions:
1.
111
2.
–10
5. 10110
1011
3.
–11
6.
1001
11011
4.
–1010
7.
1001
110
–1
8.
10110
–1001
–110
–111
–1101
9. 11011
10. 11001
11. 101010
12. 100000
–1010
–1110
–10101
–1111
Unit 2
Square and Square Roots
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2.4 Multiplication
When we consider multiplication in the binary scale we need to remember
only the following facts:
0 × 0
=
0
0 × 1
=
0
1 × 0
=
0
1 × 1
=
1
or
×
0
1
0
0
0
1
0
1
Examples (all ‘long’ multiplication).
(i)
101
check
5
(ii)
101
check
5
11
×3
101
×5
101
15
101
25
101
101
1111
11001
Exercise 2.5
1.
1011 × 11
2. 1101 × 10
3. 1001 × 110
4.
1011 × 1001
5. 1111 × 1010
6. 10101 × 1100
2.5 Division
Examples
(essentially long division).
101
(i)
1011 ÷10two
5
10)1011
check
10
10
11
1
10
1
14
Unit 2
Square and Square Roots
2) 11
remainder
11
(ii)
1011 ÷ 11 two
3
11)1011
check
3)11
11
9
101
2
11
10
remainder
110
6
(iii) 100000 ÷ 101two 101)100000
check
101
5)32
30
110
2
101
10
remainder
110
(iv) 11110 ÷ 101two
6
101)11110
check
101
5)30
30
101
101
Exercise 2.6
1.
100011 ÷ 101
2.
100011 ÷ 111
3.
1010100 ÷ 1100
4.
1010100 ÷ 111
5.
10011001 ÷ 10001
6.
110001 ÷ 111
Unit 2
Square and Square Roots
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