Class 7 Maths Notes for Exponents And Powers

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CLASS 7
MATHS

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EXPONENTS AND POWERS

I. Important Concepts: - The continued product of a number multiplied with itself a number
of times can be written as the number raised to the power a natural number, equal to the
number of times the number is multiplied with itself.
Ex: 5 × 5 × 5 can be written as 53 and it read as 5 raised to the power 3 or third power of 5
Similarly, a × a = a2 , a × a × a = a3 , a × a × a × a = a4
In general, if n is a natural number, then,
a × a × a × a × ………... × a = an (an is called nth power of a)
Laws of exponents:
(1) am × an = am+n 33 × 37 = 33+7 = 310
(2) am ÷ an = am-n 57 ÷ 53 = 57-3 = 54
(3) (am)n = am×n = (an)m (43)5 = 43×5 = 415
(4) an × bn = (ab)n 34 × 24 = (3×2)4 = 64
(5) a0 = 1 10000 = 1
1 1 1 1
(6) a-m = a m 2-5 = 25 = 2 ×2 ×2 ×2 ×2 = 32

Expanded Form:8604 = 8 × 103 + 6 × 102 + 0 × 101 + 4 × 100
Standard Form: 27065 = 2.7065 × 104; 0.3567 = 3.567 × 10-2; 0.000056 = 5.6 × 10-5
General Form: 3.5 × 103 = 3500; 3.76 × 10-6 = 0.00000376.

II. MCQs.
1) Write exponential form for 8 × 8 × 8 taking base as 2.
a) 28 b) 210 c) 212 d) 24
81
2) Express 256 in exponential form.

a)
3 3 4 3 3 d) 3
4
4 b) c)
4 4

2 6
3) The reciprocal of 5 is.

2 6 b) 26 c) 56 5 6
a) d)
5 2

4) Is 23 is equal to 32.
a) 32> 23 b) 3>2 c) 23> 32 d) 0

5) Simplify 23 × 7.
6) Evaluate (-4)4.
7) Write 4963 in expanded form.
8) Find the value of (5x)3 when x = -2.

96

9) Simplify (30 + 4-1) × 23.
(n 2 ×3n)
10) The number of diagonals of an n-sides polygon is . Use the formula to find.
2

a) Number of diagonals of triangle.
b) Number of diagonals of hexagon.
c) Number of diagonals of octagon.

Questions for Practice.
MCQ
11) Find the value of 73.
a) 10 b) 21 c) 49 d) 343
12) Express in exponential form 2 × 2 × a × a × a × a.
a) (2a)2 b) (2a)4 c) 22 × a4 d) None of these
13) Express in exponential form 625
a) 252 b) 54 c) 53 d) None of these
14) Which is greater?
a) 34> 43 b) 4>3 c) 43 > 34 d) 0
15) Simplify: 83 × 33
a) (24)6 b) (24)9 c) 5 d) (24)3
16) Simplify: (-2)5
a) (-32) b) 32 c) (-10) d) None of
these
17) Simplify: 20 × 50 × 70
a) 2 b) 20 c) 1 d) 3
1 −3
18) The value of 8 is

a) 512 b) 24 c) 5 d) 11
19) Write 5900 in standard form.
a) 59 b) 5.9 × 103 c) 5.90 d) 59 × 100
20) The value of (-1)odd number is:

a) 1 b) 2 c) -1 d) 0

97

III. Short Answer Type.
1) Express as product of power 540.
2) Simplify (-3)2 × 23.
3) Simplify 33 × 103.
4) Simplify 82 ÷ 23.
5) Find (20 × 30 + 40)
6) Find 40 – 20.
7) Evaluate 23 × a3 × b3.
8) Write 318000000 in standard form.
9) Write 0.00000898 in standard form.
10) Write 3 × 103 + 4 × 102 + 5 × 100 in usual form.
IV. Long Answer Type.
1) Find the value of (-7)3 × (-10)3.
2) Simplify 38 ÷ 32 × 34.
3) Simplify [ (32)2 × 36] × 3-8.
4) Simplify (2 × 34 × 25) ÷ (9 × 42)
5) Evaluate 34 × a6 × 5a4.
6) Find the value of (70 – 9-1) × 33.

Case Study:
7. In our earth 36141900 square Km of area is covered with water and 148647000 square Km of
area is covered with land. Find
a) Area of water in standard form?
b) Area of land in standard form?
c) Which has greater area?
8. A teacher shows 4 articles of different lengths (ball, cube, basket & book) in a classroom.
The difficulty is that the length are in exponential form. The length of different articles are as
following
i. 3 ×163/4 ii. 41/2 × 272/3 iii. 70 × 813/4 iv. 6 × 93/2 × 3-2

Then, find.
1) The length of ball is
a) 6 b) 12 c) 24 d) None of
these
2) The product of length I ball, and III basket is
a) 192 b) 144 c) 324 d) 648
st th
3) The ratio of the length of 1 and 4 articles is

98

a) 3:4 b) 4:3 c) 1:1 d) 4:9

Solution:
II. MCQ.
1) 212→ c
3 4
2) →b
4
5 6
3) →d
2
4) 3 > 23→ a
2

5) 23 × 7 = 2 × 2 × 2 × 7 = 8 × 7 = 56.
6) (-4)4 = (-4) × (-4) × (-4) × (-4) = 16 × 16 = 256.
7) 4963 = 4 × 1000 + 9 × 100 + 6 × 10 + 3 × 100 = 4 × 103 + 9 × 102 + 6 × 101 + 3 × 100.
8) (5x)3 = [5 × (-2)]3 = 53 × (-2)3 = 125 × (-8) = -1000.
1 4+1 5
9) (30 + 4-1) × 23 = 1 + 4 × 23 = × 2 × 2 × 2 = 4 × 8 = 10.
4
10) (a) Zero
(n 2 −3n) (62 − 3 × 6) (36−18)
(b) n = 6, No. of diagonals = = = = 9.
2 2 2
(n 2 −3n) (82 − 3 × 8) (64−24) 40
(c) n = 8, No. of diagonals = = = = 2 = 20.
2 2 2
III. MCQ.
1) 73 = 7 × 7 × 7 = 343 → d 6) (-32) → a
2) 22 × a4 → c 7) 1 → c
3) 54 → b 8) 512 → a
4) 34> 43 → a 9) 5.9 × 103 → b
5) (24)3→ d 10) -1→ c

IV. Short Answer Type.
1) 540 = 2 × 2 × 3 × 3 × 5 = 22 × 32 × 5
2) (-3)2 × 23 = (-3) × (-3) × 2 × 2 × 2 = 9 × 8 = 72
3) 33 × 103 = (3 × 10)3 = 30 × 30 × 30 = 27000
4) 82 ÷ 23 = (23)2 ÷ 23 = 26 ÷ 23 = 26-3 = 23 = 8
5) (20 × 30 + 40) = 1 × 1 + 1 = 1 + 1 = 2
6) 40 – 20 = 1 – 1 = 0
7) 23 × a3 × b3 = (2 × a × b)3 = (2ab)3
8) 318000000 = 3.18 × 108
9) 0.00000898 = 8.98 × 10-6
10) 3 × 103 + 4 × 102 + 5 × 100 = 3000 + 400 + 5 = 3405.
V. Long Answer Type:
1) (-7)3 × (-10)3 = (-7) × (-7) × (-7) × (-10) × (-10) × (-10) = 343000
2) 38 ÷ 32 × 34 = 38 ÷ 32+4 = 38 ÷ 36 = 38-6 = 32 = 3 × 3 = 9
3) [ (32)2 × 36] × 3-8 = [ 32*2 × 36] × 3-8 = [34 × 36] × 3-8 = 34+6 × 3-8 = 310-8 = 32 = 9

99

2 × 34 × 25
4) = 2 × 34 × 25 × 3-2 × 2-4 = 26 × 2-4 × 34 × 3-2 = 22 × 32 = 4 × 9 = 36
32 × 24
5) 34 × a6 × 5a4 = (3 × 3 × 3 × 3 × 5) × a6 × a4 = 805a10.
1 (9−1)
6) (70 – 9-1) × 33 = 1 − 9 × 3 × 3 × 3 = × 3 × 3 × 3 = 8 × 3 = 24.
9
7) Case Study.
a) Area of water = 36141900 = 3.61419 × 108 square Km.
b) Area of land = 148647000 = 1.48647 × 108 square Km.
c) Area of water is greater.
8) a) 24 → c b) 648 → d c) 4:3 → b

Chapter Test – I

Time: 1 Hour M.M.: 20

Note: All questions are compulsory.
Section A (1 mark each)
1. The number 243 can be expressed in the exponential notations as:
a) 34 b) 35 c) 33 d) 93

2. The standard form of 85,00,000 is.
a) 8.5 × 107 b) 8.5 × 105 c) 8.0 × 107 d) None of these

3. The value of 153 is.
a) 2275 b) 3315 c) 3335 d) 3375

4. Simplify: 54 × 24
a) 100 b) 400 c) 1000 d) None of these

Section B (2 marks each)
2/5
5. The value of n in n = 32 is
6. Simplify (-2)-5 × (-1)-5.
7. Express in exponential form 2187.

Section C (3 marks each)
4 3 2
8. Simplify (4 × 3 × 4 ) ÷ (27 × 4 ).
9. Express in standard form 3,40,30,000.

Section D (4 mark)

100

10. Simplify (25 × 23 × t8) ÷ (103 × t4)
Or
Express as a product of prime factors only in exponential form 729 × 64.
Chapter Test – II
Time: 1½ Hours M.M.: 30

Note: All questions are compulsory.
Section A (1 mark each)
0 0 0
1. The value of 2 + 3 + 4 is:
a) 9 b) 24 c) 3 d) None of these
2. Write the exponential form of 270.
a) 2 × 3 × 53 b) 22 × 5 × 32 c) 2 × 52 × 32 d) None of these
3. Simplify 23 × a3 × 5 × a5.
a) 40a15 b) 40a8 c) 10a8 d) 40a2
4. Simplify: 82 ÷ 43
a) 1 b) 4 c) 8 d) 12
5. Write 162 as a power of 2 is:
a) 26 b) 28 c) 24 d) 216
6. Simplify (32)4
a) 32 b) 36 c) 30 d) 38

Section B (2 marks each)
7. Express 144 as the product of power of prime numbers.
8. Write 43010000 in standard form.
2 3
9. Evaluate − 5
10. Simplify (25 × 34) ÷ 64.
11. Simplify (-2)5 × (-10)2.

Section C (3 marks each)
21 4 19 4
12. Simplify 19 × 7
0 1/3
13. Simplify 3 – 8
14. If x=3, find the value of 271/x

Section D (5 mark)
15. In our earth, 361419000 square km of area is covered with water and 148647000 square km
of area is covered with land. Find
a. Area of water in standard form. (2)

101

b. Area of land in standard form. (2)
c. Which has greater area. (1)

Solution: Test I
1) 35→ b 2) 8.5 × 107→ a 3) 3375 → d 4) 1000 → c
2/5 5 * 2/5 2
5) n = 32 = 2 = 2 = 2 × 2 = 4.
1 1 1
6) (-2)-5 × (-1)-5 = (−2)5 × −1 5 = −32 ×(−1) = 32
7) 2187 = 3 × 3 × 3 × 3 × 3 × 3 × 3 = 37.
4 × 34 ×43
8) = 43+1 × 34 × 3-3 × 4-2 = 44-2 × 34-3 = 42 × 3 = 48.
27 × 4 2
9) 3,40,30,000 = 3.43 × 106.
(25 × 23 × t 8 ) 52 × 23 × t 8 t4
10) = 53 × 23 × t 4 = 5 .
10 3 × t 4
Or
729 × 64 = 3 × 3 × 3 × 3 × 3 ×3 × 2 × 2 × 2 × 2 × 2 × 2 = (3 × 2)6 = 66.
Solution: Test II
1) 3 → c 2) 2 × 3 × 53 → a 3) 40a8→ b 4) 1 → a 5) 28→ b 6) 38→ d
7) 144 = 2 × 2 × 2 × 2 × 3 × 3 = 24 × 32.
8) 43010000 = 4.301 × 107.
−2 3 −2 × −2 ×(−2) −8
9) = = 125 .
5 5 ×5 × 5
10) (25 × 34) ÷ 64 = (25 × 34) ÷ (24 × 34) = 25 × 2-4 × 34 × 3-4 = 2.
11) (-2)5 × (-10)2 = (-32) × 100 = -3200.
21 4 19 4 74 × 34
12) × = = 3 × 3 × 3 × 3 = 81.
19 7 74
0 1/3 3 × 1/3
13) 3 – 8 = 1 – 2 = 1 – 2 = 1.
1/x 3 × 1/3
14) 27 = 3 = 3.
15)
a. Area of Water = 361419000 = 3.61419 × 108 square km.
b. Area of land = 148647000 = 1.48647 × 108 square km.
c. Water has greater area.

102

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