Class 7th | Exponents and Power |
Exponents and Power
Numbers in exponential form obey certain laws, which are:
For any non-zero integers a and b and whole numbers m and n,
(a) a ^ prime prime * a^ prime prime =a^ prime prime +n
(b) (a ^ m) / (a ^ n) = a ^ (m - n) m > n
(c) (a ^ m) ^ n = a ^ (mn)
(d) a^ prime prime prime * b^ prime prime = (ab) ^ m
(e) (a ^ m) / (b ^ m) = (a/b) ^ m
(f) a ^ 0 = 1
(g) (-1) even number = 1 (-1)odd number 1
1.1 Base and Power
= A × A × A × A × A
= A⁵
The short notation of A × A × A × A × A is A⁵
Here ‘A’ is called the base and ‘5’ is called the exponent.
In opposite
A⁵ = A × A × A × A × A
Here A⁵ is called written in exponential form and A × A × A × A × A is called written in expanded form.
Now,
A²
The whole A² is written in exponential form.
7In A² the Digit A is called Base and Digit ² is called Power or Exponent.
Observe 1,00, 000
= 10 × 10 × 10 × 10 × 10
= 10⁵
The short notation of 1,00,000 is 10⁴
Here ‘10’ is called the base and ‘5’ is called the exponent.
1.2 Reading of Power
Instead of taking a fixed number let us take any integer a as the base, and write the
numbers as,
a = a¹
(read as ‘a')
a × a = a²
(read as ‘a squared’ or ‘a raised to the power 2’)
a × a × a = a³
(read as ‘a cubed’ or ‘a raised to the power 3’)
a × a × a × a = a⁴
(read as a raised to the power 4 or the 4th power of a)
..............................
a × a × a × a × a × a × a = a⁷
(read as a raised to the power 7 or the 7th power of a)
and so on.
a × a × a × b × b can be expressed as a³ b²
(read as a cubed b squared)
EXAMPLE 6 Work out (1)⁵, (–1)³, (–1)⁴, (–10)³, (–5)⁴
.
SOLUTION
(i) We have (1)⁵
= 1 × 1 × 1 × 1 × 1
= 1
In fact, you will realise that 1 raised to any power is 1.
(ii) (–1)³
= (–1) × (–1) × (–1)
= 1 × (–1)
= –1
(iii) (–1)⁴
= (–1) × (–1) × (–1) × (–1)
= 1 ×1
= 1
You may check that (–1) raised to any odd power is (–1), and (–1) raised to any even power is (+1).
(iv) (–10)³
= (–10) × (–10) × (–10)
= 100 × (–10)
= – 1000
(v) (–5)⁴
= (–5) × (–5) × (–5) × (–5)
= 25 × 25
= 625
01. What is the base and exponent of the followings:
(i) 2⁷
(i) 5 -²
(ii) (– 2 × 3)⁰
(iii) (7⁰)²
(i) 2⁶
(ii) 9³
(iii) 11²
(iv) 5⁴
Sol. (a)
In 2⁷
Base = 2
Power = 7
02 Express 256 as a power 2.
Sol.
We have 256
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.
= 2⁸
So we can say that 256 = 2⁸
03. Express the following numbers as a product of powers of prime factors:
(i) 72 (ii) 432 (iii) 1000 (iv) 16000 (v) 512 (vi) 343 (vii) 729 (viii) 3125
Sol. (i)
(i) 72
= 72
= 2 × 2 × 2 × 3 × 3
= 2³ × 3²
Thus, 72 = 2³ × 3²
(required prime factor product form)
04. Express:
(i) 729 as a power of 3
(ii) 128 as a power of 2
(iii) 343 as a power of 7
05. Express the following in exponential form:
(A)
(a) 2 × 2
(b) 3 × 3
(c) 5 × 5
(d) 5 × 5 × 5
(e) 6 × 6 × 6 × 6
(f) 7 × 7 × 7 × 7 × 7
(g) (–3)(–3)
(h) (–2)(–2)(–2)
(i) (–3)(–3)(–3)(–3)
(j) 6 × 6 × 6 × 6
(k) 5 × 5 × 7 × 7 × 7
(l) (–5)(–5)(–5)(–5)(–5)
(m) 6 × 6 × 6 × 6 × 6 × 6
(n) 5 × 5 × 7 × 7 × 7
(B)
(a) t × t
(b) x × x
(c) a × a × a
(d) m × m × m
(e) b × b × b × b
(f) n × n × n × n × n
(g) b × b × b × a × a
(h) b × b × a × a × a
(i) a × a × b × b × b
(j) a × a × a × a × b
(k) b × b × b × b × b × a × a × a
(l) a × a × a × c × c × c × c × d
(C)
(a) 2 × t
(b) x × 3
(c) 2 × a × a
(d) 5 × m × m
(e) 2 × 3 × b × b
(f) 2 × 2 × n × n × n
(g) 3 × 5 × b × a × a × b × b × 2
(h) 2 × 5 × a × a × a
(i) 1 × 2 × b × b × b
(j) a × a × 1 × 2 × 3
(k) 2 × b × b × b × 5 × a × a × a
(l) 2 × a × a × 7 × c × c × c × d
(m) 2 × 2 × a × a
(n) 4 × 4 × a × a × b × b × b
06. Express 64 × 27 in exponential form.
07. Express (– 27/125) in exponential form.
08. Express 128/81 in exponential form.
6. Simplify:
(a) 2 × 10³
(b) 7² × 2²
(c) 2³ × 5
(d) 3 × 4⁴
(e) 0 × 10²
(f) 5² × 3³
(g) 2⁴ × 3²
(h) 3² × 10⁴
(i) (– 4)³
(j) (–3) × (–2)³
(k) (–3)² × (–5)²
(l) (–2)³ × (–10)³
8. Compare the following numbers:
(i) 2.7 × 10¹² ; 1.5 × 10⁸
(ii) 4 × 10¹⁴ ; 3 × 10¹⁷
Exercise 02
Simplyfy and write in exponential form:
Or
Express the following terms in the exponential form:
(i) 2⁵ × 2³
(ii) p³ × p²
(iii) 4² × 4³
(iv) a³ × a² × a⁷
(v) 5³ × 5⁷ × 5¹²
(vi) (– 4)¹⁰⁰ × (–4)²⁰
(i) 11⁶ ÷ 11²
(ii) 10⁸ ÷ 10⁴
(i) 2⁹ ÷ 2³
(iii) 9¹¹ ÷ 9⁷
(iv) 20¹⁵ ÷ 20¹³
(v) 7¹³ ÷ 7¹⁰
(a) (2³)²
(b) (3²)⁴
(c) (b²)⁴
(d )(a²)³
(e) (aⁿ)³
(a) (2³)²
(b) (4²)⁶
(c) (2²)¹⁰⁰
(d )(7⁵⁰)²
(e) (5³)⁷
(i) 4³ × 2³
(ii) 2⁵ × b⁵
(iii) a² × t²
(iv) 5⁶ × (–2)⁶
(v) (–2)⁴ × (–3)⁴
(i) 4⁵ ÷ 3⁵
(ii) 2⁵ ÷ b⁵
(iii) (– 2)³ ÷ b³
(iv) p⁴ ÷ q⁴
(v) 5⁶ ÷ (–2)⁶
3⁵ ÷ 3⁵ = 3⁵ - ⁵ = 3⁰ = 1
a⁵ ÷ a⁵ = a⁵ - ⁵ = a⁰ = 1
m⁴ ÷ m⁴ = m⁴ - ⁴ = m⁰ = 1
(i) (2 × 3)⁵
(ii) (2a)⁴
(iii) (– 4m)³
Using laws of exponents, simplify and write the answer in exponential form:
(i) 3² × 3⁴ × 3⁸
(ii) 6¹⁵ ÷ 6¹⁰
(iii) a³ × a²
(iv) 7ⁿ ×7²
(v) (5² )³ ÷ 5³
(vi) 2⁵ × 5⁵
(vii) a⁴ × b⁴
(viii) (3⁴ )³
(ix) (2²⁰ ÷ 2¹⁵) × 2³
(x) 8ⁿ ÷ 8²
11Simplify and write the answer in the exponential form.
(i) (3⁷/3²)×3³
(ii) 2³ × 2² × 5⁵
(iii) (6² × 6⁴) ÷ 6⁴
(iv) [(2²)³ × 3⁶] × 5⁶
(v) 8² ÷ 2³
12. Simplify:
(ii) 2³ × a³ × 5a⁴
(ii) 25⁴ ÷ 5³
(ii) {(5²)³ × 5³} ÷ 5⁷
(ii) (2³ × 2)²
(ii) (a⁵ / a³) × a⁸
(i) 12⁴ × 9³ × 4
6³ × 8³ × 27
(ii) 2³ × a³ × 5a⁴
(iii) 2 × 3⁴ × 2⁵
9 × 4²
(iii) 2³ × 3⁴ × 8
9 × 32
(iii) 2³ × 3⁴ × 4
3 × 32
(iii) 3 × 7² × 11⁸
21 × 11³
(iii) 3⁷
3⁴ × 3³
(iii) 2⁸ × a⁵
4³ × a³
(iii) 4⁵ × a⁸b³
4⁵ × a⁵b²
(iii) (2⁵)² × 7³
8³ × 7
(iii) 25 × 5² × t⁸
10³ × t⁴
(iii) 3⁵ × 10⁵ × 25
5⁷ × 6⁵
4. Express each of the following as a product of prime factors only in exponential form:
(i) 108 × 192
(ii) 270
(iii) 729 × 64
(iv) 768
01. Find the value of
(i) 7³
(ii) (– 3)⁵
(iii) 2⁴
(i) 5 -²
(ii) (– 2 × 3)⁰
(iii) (7⁰)²
(i) 5 × 10⁴
(ii) (–3)³ × (–10)³
02. Find the value of (– 3)² ×(– 3)⁴
03. Find the value of 2³ × 3³
04. Find the value of (2 -¹ × 3 -¹ × 4 -¹)²
05. Find the value of (– 1)¹⁰ + ( –1)¹⁰¹ + (–1)⁵¹
06. Find the value of (1/36)³ × (– 6)³
07. Find the value of (a/b)-¹ × (b/a)² × a³ × b³
08. Find the value of (a⁰ + b⁰)(a⁰ – b⁰)
09. Find the value of (7⁰ + 5⁰)(8⁰ – 3⁰)
10. Find the value of [(2p³)³] / 3 × (p²)³
11. Find value of (81)-½ × (25)½
12. Find the value of [(36)½ + (16)½ × 1]²
13. Find the value of (–1/64)³/²
14. Find the value of
[(2/5)⁵ × (2/5) ²]
[4/9 × (2/5)³]
Exercise 03
01. Find the value of n:
(i) 1 million = 10ⁿ
(ii) 1 lakh = 10ⁿ
(iii) 2⁵ × 4² = 2ⁿ
(iv) (100²) / (10⁴) = 10ⁿ
02. Find the value of ⁿ if (3⁴)⁵ = 3²ⁿ
03. If 4²ⁿ = 64 then find the value of n.
04. Find the value of n if 3²ⁿ × 3² = 3⁸
05. If [(2/5)² × (2/5)ⁿ] = 1 then find the value of n.
06. If 2²ⁿ - ³ = 64ⁿ find the value of n.
07. Find the value of n if [(–8/3)¹⁰] ÷ [(–8/3)⁴] = (–8/3) ²ⁿ + ²
08. Find the value of n if, ((2⁶) / (2-³)) × 2¹⁴ = 2ⁿ
09. Find the value of (1/4)-² + (1/2)-² + (1/3)-²
10. If a = 2 and b = 3 find the value of a² + b³
11. Find the value of (3-² – 5-¹) × 13⁰.
12. Find the value of and express in exponential form:
{(2⁵)² × 3⁴} ÷ {2⁸ × 3²}
13. Find the value and express in exponential form [(6⁷)/(6²)] × 6⁵
14. Find the value of and express in exponential form (5⁴ × 5⁶) / (5¹⁰)
Exercise 04
01. Which one is greater 2³ or 3²?
02. Which one is greater 8² or 2⁸ ?
03. Which one is greater (2²)×5 or (2²)⁵ ?
04. Can you tell which one is greater (5²) × 3 or (5² )³ ?
05. Identify the greater number, wherever possible, in each of the following?
(i) 4³ or 3⁴
(ii) 5³ or 3⁵
(iii) 2⁸ or 8²
(iv) 100² or 2100
(v) 2¹⁰ or 10²
Sol. 01 We have,
2³
= 2 × 2 × 2
= 8 and
3²
= 3 × 3
= 9.
Since 9 > 8,
so, 3² is greater than 2³
Sol. 02
8²
= 8 × 8
= 64
2⁸
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= 256
Clearly, 2⁸ > 8²
Exercise 04
01. Find the product of square of 2 and square of 3.
02. Express (- 3/2)³ in the form p/q
03. If 2 ^ x × 3 ^ y = 64 × 81 , then find the value of x + y
04. If p/q = (2/7)² – (2/7)⁰ then find the value of q/p.
05. Express 4 × 4 × 4 taking base as 2.
32. Change (2/3)-³ in to positive exponent.
33. Change (3²)⁵ in to negative exponent.
Exercise 05
DECIMAL NUMBER SYSTEM
(Part 01)
we already know that
1 = 10⁰
10 = 10¹
100 = 10²
1000 = 10³
10,000 = 10⁴ etc.
Let us look at the expansion of 87561, which we already know that
87561 = 8 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1
We can express it using powers of 10 in the exponent form:
Therefore,
87561 = 4 × 10⁴ + 7 × 10³ + 5 × 10⁴ + 6 × 10¹ + 1 × 10⁰
(Note 10,000 = 10⁴, 1000 = 10³, 100 = 10², 10 = 10¹ and 1 = 10⁰ )
Let us expand another number:
204378 = 1 × 100,000 + 0 × 10,000 + 4 × 1000 + 2 × 100 + 7 × 10 + 8 × 1
= 2 × 10⁵ + 0 × 10⁴ + 4 × 10³ + 3 × 10² + 7 × 10¹ + 8 × 10⁰
= 2 × 10⁵ + 4 × 10³ + 3 × 10² + 7 × 10¹ + 8 × 10⁰
Notice how the exponents of 10 start from a maximum value of 5 and go on decreasing
by 1 at a step from the left to the right upto 0.
(Part 02)
We know that each digit is followed by a decimal point (.), whether it is written in the decimal place or not.
6 = 6.0
6.9 = 6.9
69 = 69.0
690 = 690.0
6900 = 6900.0
69000 =69000.0
690000 =690000.0
690000 =6900000.0
These numbers are not convenient to write and read. To make it convenient we use powers.
Observe the following:
69 = 6.9 × 10 = 6.9 × 10¹
690 = 6.9 × 100 = 6.9 × 10²
6900 = 6.9 × 1000 = 6.9 × 10³
69000 =6.9 × 10000 = 6.9 × 10⁴
690000 =6.9 × 100000 = 6.9 × 10⁵
690000 =6.9 × 1000000 = 6.9 × 10⁶
and so on.
EXPRESSING LARGE NUMBERS IN THE STANDARD FORM
The large numbers are not convenient to write and read. To make it convenient we use powers. We said that large numbers can be
conveniently expressed using exponents.
Standard Form or Scientific Notation Of The Number
We know that any number can be expressed as a decimal number. Standard Form Of The Number is also known as Scientific Notation of the number.
The number start as 1.0 and multiplied by a power of 10 is called the Standard Form Of The Number.
Such a form of a number A.B × 10ⁿ is called its standard form. (Where A, B and n are numbers)
★ The, distance of Sun from the centre of our Galaxy i.e., 300,000,000,000,000,000,000 m can be written as
= 3.0 × 100,000,000,000,000,000,000
= 3.0 × 10²⁰ m
★ Now, can you express 40,000,000,000 in the similar way?
Count the number of zeros in it. It is 10.
So,
= 40,000,000,000
= 4.0 × 10,000,000,000
= 4.0 × 10¹⁰
★ Mass of the Earth
= 5,976,000,000,000,000,000,000,000 kg
= 5.976 × 10²⁴ kg
Mass of Uranus
= 86,800,000,000,000,000,000,000,000 kg
= 8.68 × 10²⁵ kg
Exercise 01
01 Expand by expressing powers of 10 in the exponential form:
(i) 172
(ii) 5,643
(iii) 56,439
(iv) 1,76,428
(v) 5985.3
(vi) 65,950
(vii) 3,430,000
(viii) 70,040,000,000
02. What number is obtained from expanded form 5 × 10⁴ + 2 × 10³ + 5 × 10² + 2 × 10¹ + 10⁰
03. Express the following numbers in the standard form:
(i) 172
(ii) 5,643
(iii) 56,439
(iv) 1,76,428
(v) 5985.3
(vi) 65,950
(vii) 3,430,000
(viii) 70,040,000,000
SOLUTION
(i) 5985.3 = 5.9853 × 1000 = 5.9853 × 10³
(ii) 65,950 = 6.595 × 10,000 = 6.595 × 10⁴
(iii) 3,430,000 = 3.43 × 1,000,000 = 3.43 × 10⁶
(iv) 70,040,000,000 = 7.004 × 10,000,000,000 = 7.004 × 10¹⁰
04. Express 2000000 in standard form.
05. The speed of light is 300000000 m/s. Express it in standard form.
06. What is the standard form of the number 2467895?
07. What is the standard form of the number 0.0032456?
Exercise 02
8. What is usual form of 3.469 × 10⁶ ?
29. What is usual form of 2.29 × 10⁴ ?
Write in the usual from
(i) 5.9853 × 10³
(ii) 6.595 × 10⁴
(iii) A= 3.43 × 10⁶
(iv) 70,040,000,000 = 7.004 × 10,000,000,000 = 7.004 × 10¹⁰
47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1
= 4 × 10⁴ + 7 × 10³ + 5 × 10² + 6 × 10¹ + 1 × 10⁰
(Note 10,000 = 10⁴, 1000 = 10³, 100 = 10², 10 = 10¹ and 1 = 10⁰
1. Write the following numbers in the expanded forms:
279404, 3006194, 2806196, 120719, 20068
2. Find the number from each of the following expanded forms:
Or
Write the following expanded forms in actual form:
(a) 8x10⁴ + 6x10³ + 0x10² + 4×10¹ + 5x10⁰
(b) 4x10⁵ + 5×10³ + 3×10² + 2×10° + 2×10⁰
(c) 3x10³ + 7×10² + 5×10⁰
(d) 9x10⁵ + 2×10² + 3×10¹
3. Express the following numbers in standard form:
(i) 5,00,00,000
(ii) 70,00,000
(iii) 3,18,65,00,000
(iv) 3,90,878
(v) 39087.8
(vi) 3908.78
4. Express the number appearing in the following statements in standard form.
(a) The distance between Earth and Moon is 384,000,000 m.
(b) Speed of light in vacuum is 300,000,000 m/s.
(c) Diameter of the Earth is 1,27,56,000 m.
(d) Diameter of the Sun is 1,400,000,000 m.
(e) In a galaxy there are on an average 100,000,000,000 stars.
(f) The universe is estimated to be about 12,000,000,000 years old.
n
(g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be 300,000,000,000,000,000,000 m.
(h) 60,230,000,000,000,000,000,000 molecules are contained in a drop of water weighing 1.8 gm.
(i) The earth has 1,353,000,000 cubic km of sea water.
(j) The population of India was about 1,027,000,000 in March, 2001.
Exponents and Power
Questions:
1. Express in exponential form:
(i) 5 × 5 × 5
(ii) (- 3)(- 3)(- 3)(- 3)
(iii) 4 × 4×a×a×b×b×b
2. Find the value of
(i) 7³
(ii) (– 3)⁵
(iii) 2⁴
3. What is the base and exponent of
(i) 5 -²
(ii) (– 2 × 3)⁰
(iii) (7⁰)²
4. Find the value of n:
(i) 1 million = 10ⁿ
(ii) 1 lakh = 10ⁿ
5. Find the value of (– 3)² ×(– 3)⁴
6. Express 64 × 27 in exponential form.
7. Express 2000000 in standard form.
8. What is usual form of 3.469 × 10⁶ ?
9. Find the value of (2 -¹ × 3 -¹ × 4 -¹)²
10. If a = 2 and b = 3 find the value of a² + b²
11. Find the value of (– 1)¹⁰ + ( –1)¹⁰¹ + (–1)⁵¹
12. Which one is greater (2²)×5 or (2²)⁵ ?
13. Find the value of (a⁰ + b⁰)(a⁰ – b⁰)
14. Find the value of 2³ × 3³
15. Find the product of square of 2 and square of 3.
16. Find the value of (1/36)³ × (– 6)³
17. Express in the form p/q (- 3/2)³
18. Find the value of n if
(i) 2⁵ × 4² = 2ⁿ
(ii) (100²) / (10⁴) = 10ⁿ
19. Find the value of [(2p³)³] / 3 × (p²)³
20. Find value of (81)-½ × (25)½
21. Find the value of ⁿ if (3⁴)⁵ = 3²ⁿ
22. Find the value of [(36)½ + (16)½ × 1]²
23. Find the value of (–1/64)³/²
Page 86
24. Express (– 27/125) in exponential form.
25. Express 128/81 in exponential form.
26. Find the value of (7⁰ + 5⁰)(8⁰ – 3⁰)
27. Find the value of and express in exponential form (5⁴ × 5⁶) / (5¹⁰)
28. Find the value of (a/b)-¹ × (b/a)² × a³ × b³
29. What is usual form of 2.29 × 10⁴ ?
30. If [(2/5)² * (2/5)ⁿ] = 1 then find the value ofn.
31. The speed of light is 300000000 m/s. Express it in standard form.
32. Change (2/3)-³ in to positive exponent.
33. Change (3²)⁵ in to negative exponent.
34. If 4²ⁿ = 64 then find the value of n.
35. Find the value of n if 3²ⁿ × 3² = 3⁸
36. What number is obtained from expanded form 5 × 10⁴ + 2 × 10³ + 5 × 10² + 2 × 10¹ + 10⁰
37. If 2²ⁿ - ³ = 64ⁿ find the value of n.
38. If p/q = (2/7)² – (2/7)⁰ then find the value of q/p.
39. Find the value of (3-² – 5-¹) × 13⁰.
40. If 2 ^ x × 3 ^ y = 64 × 81 , then find the value of x + y
41. What is the standard form of the number 2467895?
42. Find the value of n if [(–8/3)¹⁰] ÷ [(–8/3)⁴] = (–8/3) ²ⁿ + ²
43. Find the value of and express in exponential form:
{(2⁵)² × 3⁴} ÷ {2⁸ * 3²}
44. What is the standard form of the number 0.0032456?
45. Express 4 × 4 × 4 taking base as 2.
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87
46. Find the value and express in exponential form ((6⁷)/(6²)) × 6⁵
47. Find the value of n if, ((2⁶) / (2-³)) × 2¹⁴ = 2ⁿ
48. Find the value of (1/4)-² + (1/2)-² + (1/3)-²
49. Find the value of:
(i) 5 × 10⁴
(ii) (–3)³ × (–10)³
50. Find the value of
[(2/5)⁵ × (2/5) ²]
[4/9 × (2/5)³]
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