CBSE Class 7 Mathematics Exponents And Powers Worksheet Set 01

1. Write each of the following in power notation:

(i) (5/7) ×(5/7) × (5/7) × (5/7)
Solution:-
The product of rational number multiplied several times by itself can be expressed in the power notations as,
(5/7)⁴

(ii) (-4/3) ×(-4/3) × (-4/3) × (-4/3) × (-4/3)
Solution:-
The product of rational number multiplied several times by itself can be expressed in the power notations as,
(-4/3)⁵

2. Express each of the following in power notation:

(i) (25/36)
Solution:-
We have,
25 = 5 × 5 = (5)²
36 = 6 × 6 = (6)²
Then,
= (5²/6²)
∴ (5/6)²

(ii)(-27/64)
Solution:-
We have,
-27 = -3 × -3 × -3 = (-3)³
64 = 4 × 4 × 4 = (4)³
Then,
= (-3³/4³)
∴ (-3/4)³

3. Express each of the following a rational number:
(i) (-8/5)³
Solution:-
We have,
(-8/5)³ = (-8³/5³)
= (-8 × -8 × -8) / (5 × 5 × 5)
= (-512/125)

(ii) (-13/11)²
Solution:-
We have,
(-13/11)² = (-13²/11²)
= (-13 × -13) / (11 × 11)
= (169/121)

4. Express each of the following as a rational number:

(i) (4)^-1
Solution:-
We have:
(4)^-1 = (4/1)^-1
= (1/4)¹ … [∵ (a/b)^-n = (b/a) ⁿ]
= (1/4)

(ii) (-6)^-1
Solution:-
We have:
(-6)^-1 = (-6/1)^-1
= (1/-6)¹ … [∵ (a/b)^-n = (b/a) ⁿ]
= (-1/6)

5. Find the reciprocal of each of the following:
(i) (3/8)⁴
Solution:-
We know that the reciprocal of (a/b) ^m is (b/a) ^m
Then,
Reciprocal of (3/8)⁴ is (8/3)⁴

(ii) (-5/6)¹¹
Solution:-
We know that the reciprocal of (a/b) ^m is (b/a) ^m
Then,
Reciprocal of (-5/6)¹¹ is (-6/5)¹¹

6. Find the value of each of the following:

(i) 8⁰
Solution:-
By definition, we have a0= 1 for every integer.
∴ 8⁰ = 1

(ii) 6⁰ × 7⁰
Solution:-
By definition, we have a⁰= 1 for every integer.
∴ 6⁰ × 7⁰
= 1 × 1
= 1

7. Simplify each of the following and express each as a rational number:

(i) (3/2)⁴ × (1/5)²
Solution:-
We have,
(3⁴/2⁴) = (3×3×3×3)/ (2×2×2×2) = (81/16)
(1²/5²) = (1×1)/ (5×5) = (1/25)
Then,
= (81/16) × (1/25)
= (81×1) / (16/25)
= (81/400)

(ii) (-2/3)⁵ × (-3/7)³
Solution:-
We have,
(-2⁵/3⁵) = (-2×-2×-2×-2×-2)/ (3×3×3×3×3) = (-32/243)
(-3³/7³) = (-3×-3×-3)/ (7×7×7) = (-27/343)
Then,
= (-32/243) × (-27/343)
= (-32×-27) / (243×343)
By simplifying,
= (-32×-1) / (9/343)
= (32/3087)

8. Simplify and express each as a rational number:

(i) (4/9)⁶× (4/9)^-4
Solution:-
We have,
= (4/9) ^{(6+ (-4))} … [{(a/b) ^m × (a/b) ⁿ} = (a/b)^m-n
= (4/9) ^(6-4)
= (4/9)²
= (4²/9²)
= (16/81)

(ii) (-7/8)^-3× (-7/8)²
Solution:-
We have,
= (-7/8) ^{(-3+ 2)} … [{(a/b) ^m × (a/b) ⁿ} = (a/b)^m-n
= (-7/8) ^(-1)
= (-8/7)

9. Express each of the following as a rational number:

(i) 5^-3
Solution:-
We know that,
= (5)^-3 = (1/5)³ … [∵ (a/b)^-n = (b/a) ⁿ]
= (1³/5³)
= (1/125)

(ii) (-2)^-5
Solution:-
We know that,
= (-2)^-5 = (-1/2)³ … [∵ (a/b)^-n = (b/a) ⁿ]
= (-1³/2³)
= (-1/8)

More QUestion

1. Find the value of the following:

a) 82
b) 63
c) 26
d) 112
e) 54
f) 25
g) (–4)³
h) (–5)⁴

2. Write in exponential form.

a) 5 × 5 × 5 × 5 × 5 = ____________
b) (–4) × (–4) × (–4) = ____________
c) 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = ____________
d) (–8) × (–8) × (–8) × (–8) = ____________
e) 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = __

Worksheet 1

Question 1. Find the value of the following:
a) \( 8^2 \)
b) \( 6^3 \)
c) \( 2^6 \)
d) \( 11^2 \)
e) \( 5^4 \)
f) \( 2^5 \)
g) \( (-4)^3 \)
h) \( (-5)^4 \)
Answer:
a) \( 8^2 = 8 \times 8 = 64 \)
b) \( 6^3 = 6 \times 6 \times 6 = 216 \)
c) \( 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \)
d) \( 11^2 = 11 \times 11 = 121 \)
e) \( 5^4 = 5 \times 5 \times 5 \times 5 = 625 \)
f) \( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \)
g) \( (-4)^3 = (-4) \times (-4) \times (-4) = -64 \)
h) \( (-5)^4 = (-5) \times (-5) \times (-5) \times (-5) = 625 \)
In simple words: Multiply the base number by itself as many times as the power tells you to.

Exam Tip: A negative base raised to an even power always gives a positive result. If raised to an odd power, it gives a negative result.

Question 2. Write in exponential form.
a) 5 × 5 × 5 × 5 × 5 = ____________
b) (–4) × (–4) × (–4) = ____________
c) 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = ____________
d) (–8) × (–8) × (–8) × (–8) = ____________
e) 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = ___________
Answer:
a) \( 5^5 \)
b) \( (-4)^3 \)
c) \( 2^8 \)
d) \( (-8)^4 \)
e) \( 3^8 \)
In simple words: Count how many times the number is multiplied. Write that count as the power on top.

Exam Tip: Keep negative numbers inside parentheses to show the power applies to both the sign and the number.

Worksheet 2

Question 1. Express each of the following numbers as a product of powers of their prime factors.
a) 36 _______________
b) 72 _______________
c) 128 _______________
d) 625 _______________
e) 343 _______________
f) 540 _______________
g) 512 _______________
h) 729 _______________
Answer:
a) \( 36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2 \)
b) \( 72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2 \)
c) \( 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7 \)
d) \( 625 = 5 \times 5 \times 5 \times 5 = 5^4 \)
e) \( 343 = 7 \times 7 \times 7 = 7^3 \)
f) \( 540 = 2 \times 2 \times 3 \times 3 \times 3 \times 5 = 2^2 \times 3^3 \times 5^1 \) (or \( 2^2 \times 3^3 \times 5 \))
g) \( 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9 \)
h) \( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \)
In simple words: Break down the number into prime factors. Then write down how many times each prime number appears.

Exam Tip: Use prime factorization (the factor tree or division method) to find the prime factors without missing any.

Question 2. Simplify.
a) 2 × 103 ________
b) 72 × 2 ________
c) 33 × 42 ________
d) 52 × 23 ________
e) 3 × 24 ________
f) 112 × 0 ________
g) 24 × 32 ________
h) 152 × 0 ________
i) (–3) × (–4)2 _______
j) 62 × 33 ________
k) (–2)4 × (–10) _______
/G) (–3)2 × (–2)4 _______
m) 28 × 0 ________
n) (–7)3 × 2 ________
Answer:
a) \( 2 \times 10^3 = 2 \times 1000 = 2000 \)
b) \( 7^2 \times 2 = 49 \times 2 = 98 \)
c) \( 3^3 \times 4^2 = 27 \times 16 = 432 \)
d) \( 5^2 \times 2^3 = 25 \times 8 = 200 \)
e) \( 3 \times 2^4 = 3 \times 16 = 48 \)
f) \( 11^2 \times 0 = 121 \times 0 = 0 \)
g) \( 2^4 \times 3^2 = 16 \times 9 = 144 \)
h) \( 15^2 \times 0 = 225 \times 0 = 0 \)
i) \( (-3) \times (-4)^2 = -3 \times 16 = -48 \)
j) \( 6^2 \times 3^3 = 36 \times 27 = 972 \)
k) \( (-2)^4 \times (-10) = 16 \times (-10) = -160 \)
l) \( (-3)^2 \times (-2)^4 = 9 \times 16 = 144 \)
m) \( 28 \times 0 = 0 \)
n) \( (-7)^3 \times 2 = -343 \times 2 = -686 \)
In simple words: First solve the exponent for each number. Then multiply the resulting numbers.

Exam Tip: Always remember that any number multiplied by zero equals zero.

Worksheet 3 - Help

Question 1. Simplify and express each of the following in exponential form using laws of exponents.
a) \( 2^2 \times 2^4 \times 2^5 \)
b) \( 7^{12} \div 7^4 \)
c) \( (4^2)^3 \)
d) \( 2^3 \times 5^3 \)
e) \( a^4 \times a^7 \)
f) \( 5^{12} \div 5^4 \)
Answer:
a) \( 2^2 \times 2^4 \times 2^5 = 2^{2+4+5} = 2^{11} \)
b) \( 7^{12} \div 7^4 = 7^{12-4} = 7^8 \)
c) \( (4^2)^3 = 4^{2 \times 3} = 4^6 \)
d) \( 2^3 \times 5^3 = (2 \times 5)^3 = 10^3 \)
e) \( a^4 \times a^7 = a^{4+7} = a^{11} \)
f) \( 5^{12} \div 5^4 = 5^{12-4} = 5^8 \)
In simple words: Use exponent laws. Add powers for multiplication of like bases. Subtract powers for division of like bases. Multiply powers for power-to-power.

Exam Tip: When multiplying terms with different bases but the same exponent, you can multiply the bases together first.

Question 2. Say true or false and justify your answer.
a) \( 3^4 \times 6^2 = 18^6 \)
c) \( 4^0 = 0 \)
d) \( 5^2 > 2^5 \)
e) \( 9^3 \times 9^6 = 9^9 \)
f) \( 2^4 \times 2^5 = 2^{20} \)
g) \( (a^x)^y = a^{xy} \)
h) \( 6^2 < 2^6 \)
Answer:
a) False. Left-hand side is \( 3^4 \times 6^2 = 81 \times 36 = 2916 \), while right-hand side is \( 18^6 = 34,012,224 \). They are not equal.
c) False. Any non-zero base raised to the power of 0 equals 1, so \( 4^0 = 1 \), not 0.
d) False. \( 5^2 = 25 \) and \( 2^5 = 32 \). Since 25 is less than 32, the statement is false.
e) True. By exponent laws, \( 9^3 \times 9^6 = 9^{3+6} = 9^9 \). This matches the right-hand side.
f) False. By exponent laws, \( 2^4 \times 2^5 = 2^{4+5} = 2^9 \), which is not equal to \( 2^{20} \).
g) True. By the power of a power law, \( (a^x)^y = a^{xy} \), which matches the right-hand side.
h) True. \( 6^2 = 36 \) and \( 2^6 = 64 \). Since 36 is less than 64, the statement is true.
In simple words: Solve both sides of each equation to see if they are equal or if the inequality is correct.

Exam Tip: Never mix up the rule for adding exponents during multiplication with multiplying exponents during a power-of-a-power operation.

Worksheet 3 - Foundation

Question 1. Simplify and write the answer in exponential form.
a) \( (4^2)^3 \div 4^4 \)
b) \( 5^{10} \times 5^4 \times 5^{18} \)
c) \( y^{25} \times y^{50} \)
d) \( 9^5 \times 10^5 \)
e) \( 15^{20} \div 15^5 \)
f) \( (5^5 \div 5^3) \times 5^1 \)
g) \( 10^0 \times 6^0 \)
h) \( 7^{18} \times 7^6 \)
Answer:
a) \( (4^2)^3 \div 4^4 = 4^6 \div 4^4 = 4^{6-4} = 4^2 \)
b) \( 5^{10} \times 5^4 \times 5^{18} = 5^{10+4+18} = 5^{32} \)
c) \( y^{25} \times y^{50} = y^{25+50} = y^{75} \)
d) \( 9^5 \times 10^5 = (9 \times 10)^5 = 90^5 \)
e) \( 15^{20} \div 15^5 = 15^{20-5} = 15^{15} \)
f) \( (5^5 \div 5^3) \times 5^1 = 5^{5-3} \times 5^1 = 5^2 \times 5^1 = 5^{2+1} = 5^3 \)
g) \( 10^0 \times 6^0 = 1 \times 1 = 1 \)
h) \( 7^{18} \times 7^6 = 7^{18+6} = 7^{24} \)
In simple words: Follow exponent rules like adding exponents during multiplication and subtracting them during division.

Exam Tip: Be sure to write the final answers in exponential form as requested, rather than calculating the big final numbers.

Question 2. Fill in the blanks.
a) 8 × 8 × 8 × 8 in exponential form with base 4 is ______.
b) 4 × 4 × 4 ×4 × 4 in exponential form with base 2 is _____.
c) 9 × 9 × 9 in exponential form with base 3 is ________.
d) 16 × 16 × 16 in exponential form with base 4 is ______.
e) 25 × 25 in exponential form with base 5 is ________.
f) 64 × 64 in exponential form with base 4 is ________.
g) 27 × 27 in exponential form with base 3 is ________.
Answer:
a) \( 8 \times 8 \times 8 \times 8 = 8^4 = (2^3)^4 = 2^{12} = (2^2)^6 = 4^6 \). So the answer is \( 4^6 \).
b) \( 4 \times 4 \times 4 \times 4 \times 4 = 4^5 = (2^2)^5 = 2^{10} \). So the answer is \( 2^{10} \).
c) \( 9 \times 9 \times 9 = 9^3 = (3^2)^3 = 3^6 \). So the answer is \( 3^6 \).
d) \( 16 \times 16 \times 16 = 16^3 = (4^2)^3 = 4^6 \). So the answer is \( 4^6 \).
e) \( 25 \times 25 = 25^2 = (5^2)^2 = 5^4 \). So the answer is \( 5^4 \).
f) \( 64 \times 64 = 64^2 = (4^3)^2 = 4^6 \). So the answer is \( 4^6 \).
g) \( 27 \times 27 = 27^2 = (3^3)^2 = 3^6 \). So the answer is \( 3^6 \).
In simple words: Convert each base number to a prime number base. Then change it to match the requested base power.

Exam Tip: Write down intermediate steps like changing bases to prime factors so you do not make simple base-conversion mistakes.