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NCERT Exemplar Solutions: Exponents & Powers | Mathematics (Maths) Class 8 PDF Download

Q: 3. What are the rules of exponents and powers?

Ans. The rules of exponents and powers include:- When multiplying numbers with the same base, add the exponents.- When dividing numbers with the same base, subtract the exponents.- When raising a power to another power, multiply the exponents.- When raising a product to an exponent, distribute the exponent to each factor.

In questions 1 to 33, out of the four options, only one is correct. Write the correct answer.
Q.1. In 2ⁿ, n is known as:
(a) Base
(b) Constant
(c) exponent
(d) Variable
Ans: c
Solution: 2 is the rational number which is the base here and n is the power of 2. Hence, it is an exponent.

Q.2. For a fixed base, if the exponent decreases by 1, the number becomes:
(a) One-tenth of the previous number.
(b) Ten times of the previous number.
(c) Hundredth of the previous number.
(d) Hundred times of the previous number.
Ans: a
Solution: Suppose for 10⁶, when the exponent is decreased by 1, it becomes 10⁵. Hence, 10⁵/10⁶ = 1/10.

Q.3. 3^-2 can be written as:
(a) 3²
(b) 1/3²
(c) 1/3^-2
(d) -2/3
Ans: b
Solution: By the law of exponent we know: a^-n = 1/aⁿ.
Hence, 3^-2=1/3²

Q.4. The value of 1/(4)^-2 is:
(a) 16
(b) 8
(c) 1/16
(d) 1/8
Ans: a
Solution: 1/(4)^-2 = 1/(1/4²) = 4² = 16

Q.5. The value of 3⁵ ÷ 3^-6 is:
(a) 3⁵
(b) 3^-6
(c) 3¹¹
(d) 3^-11
Ans: c
Solution: By the law of exponents, we know,
am/an=am-n
Hence, 3⁵ ÷ 3^-6 = 3^5-(-6) = 311

Q.6. The value of (2/5)^-2 is:
(a) 4/5
(b) 4/25
(c) 25/4
(d) 5/2
Ans: c
Solution: By the law of exponent we know: a^-n = 1/aⁿ.
Hence, (2/5)^-2 = 1/(2/5)² = 1/(4/25) = 25/4

Q.7. The reciprocal of (2/5)-1 is:
(a) 2/5
(b) 5/2
(c) –5/2
(d) –2/5
Ans: b
Solution: By the law of exponent we know: a^-n = 1/an.
Hence, (2/5)^-1=1/(2/5)=5/2

Q.8. The multiplicative inverse of 10^-100 is
(a) 10
(b) 100
(c) 10¹⁰⁰
(d) 10^-100
Ans: c
Solution: By the law of exponent we know: a^-n = 1/aⁿ.
So, 10^-100 = 1/10¹⁰⁰
The multiplicative inverse for any integer a is 1/a, such that;
a x 1/a = 1
Hence, the multiplicative inverse for 1/10¹⁰⁰ is 10¹⁰⁰
as, 1/10¹⁰⁰ x 10¹⁰⁰ = 1

Q.9. The value of (–2)^2×3-1 is
(a) 32
(b) 64
(c) – 32
(d) – 64
Ans: c
Solution: (–2)^2×3-1=(-2)^6-1=(-2)⁵=-32

10. The value of (-2/3)⁴ is equal to:
(a) 16/81
(b) 81/16
(c) -16/81
(d) 81/ −16
Ans: a
Solution: (-2/3)⁴ = (-2/3)(-2/3)(-2/3)(-2/3) = 16/81

Q.11. The multiplicative inverse of (-5/9)^-99 is:
(a) (-5/9)⁹⁹
(b) (5/9)⁹⁹
(c) (9/-5)⁹⁹
(d) (9/5)⁹⁹
Ans: a

Q.12. If x be any non-zero integer and m, n be negative integers, then x^m × xⁿ is equal to:
(a) x^m
(b) x^m+n
(c) xⁿ
(d) x^m-n
Ans: b
Solution: x^m+n (By the law of exponents)

Q.13. If y be any non-zero integer, then y0 is equal to:
(a) 1
(b) 0
(c) – 1
(d) Not defined
Ans: a
Solution: 1 (By the law of exponent)

Q.14. If x be any non-zero integer, then x-1 is equal to
(a) x
(b) 1/x
(c) – x
(c) -1/x
Ans: b
Solution: 1/x (By the law of exponents)

Q.15. If x be any integer different from zero and m be any positive integer, then x^-m is equal to:
(a) x^m
(b) –x^m
(c) 1/x^m
(d) -1/x^m
Ans: c
Solution: 1/x^m (By the law of exponents)

Q.16. If x be any integer different from zero and m, n be any integers, then (x^m)ⁿ is equal to:
(a) x^m+n
(b) x^mn
(c) x^m/n
(d) x^m-n
Ans: b
Solution: x^mn (By the law of exponents)

Q.17. Which of the following is equal to (-3/4)^-3?
(a) (3/4)^-3
(b) – (3/4)^-3
(c) (4/3)³
(d) (-4/3)³
Ans: d
Solution: (-3/4)^-3 = 1/(-3/4)³ = (-4/3)³
(By the law of exponents: a^-n = 1/aⁿ)

Q.18. (-5/7)^-5 is equal to:
(a) (5/7)^-⁵
(b) (5/7)⁵
(c) (7/5)⁵
(d) (-7/5)⁵
Ans: d
Solution: (-5/7)^-5=1/(-5/7)⁵=(-7/5)⁵
(By the law of exponents: a^-n = 1/aⁿ)

Q.19. (-7/5)^-1 is equal to:
(a) 5/7
(b) – 5/7
(c) 7/5
(d) -7/5
Ans: b
Solution: (-7/5)^-1= 1/(-7/5) = -5/7

Q.20. (–9)³ ÷ (–9)⁸ is equal to:
(a) (9)⁵
(b) (9)^-5
(c) (– 9)⁵
(d) (– 9)^-5
Ans: d
Solution: (–9)³ ÷ (–9)⁸ = (-9)^3-8= (-9)^-5
(By the law of exponents: am ÷ an=am-n)

Q.21. For a non-zero integer x, x⁷ ÷ x¹² is equal to:
(a) x⁵
(b) x¹⁹
(c) x^-5
(d) x^-19
Ans: c
Solution: x⁷ ÷ x¹² = x^7-12 = x^-5
(By the law of exponents: am ÷ an=am-n)

Q.22. For a non-zero integer x, (x⁴)^-3 is equal to:
(a) x¹²
(b) x^-12
(c) x⁶⁴
(d) x^-64
Ans: b
Solution: (x⁴)^-3 = x⁴^×(-3) = x^-12
(By the law of exponents: (a^m)ⁿ=a^mn)

Q.23. The value of (7^-1 – 8^-1)^-1 – (3^-1 – 4^-1)^-1 is:
(a) 44
(b) 56
(c) 68
(d) 12
Ans: a
Solution: (7^-1 – 8-1)^-1 – (3^-1 – 4^-1)^-1
= (1/7-1/8)^-1 – (1/3-1/4)^-1
= (1/56)^-1 – (1/12)^-1
= 56 – 12 = 44

Q.24. The standard form for 0.000064 is
(a) 64 × 10⁴
(b) 64 × 10^-4
(c) 6.4 × 10⁵
(d) 6.4 × 10^-5
Ans: d

Q.25. The standard form for 234000000 is
(a) 2.34 × 10⁸
(b) 0.234 × 10⁹
(c) 2.34 × 10^-8
(d) 0.234 × 10^-9
Ans: a
Solution: 234000000 = 234 × 10⁶ = 2.34 × 10² × 10⁶ = 2.34 × 10⁸

Q.26. The usual form for 2.03 × 10^-5
(a) 0.203
(b) 0.00203
(c) 203000
(d) 0.0000203
Ans: d

Q.27. (1/10)⁰ is equal to
(a) 0
(b) 1/10
(c) 1
(d) 10
Ans: c
Solution: 1 Since, a⁰ = 1 (by law of exponent)

Q.28. (3/4)⁵ ÷(5/3)⁵ is equal to
(a) (3/4÷5/3)⁵
(b) (3/4 ÷ 5/3)¹
(c) (3/4 ÷ 5/3)⁰
(d) (3/4 ÷ 5/3)¹⁰
Ans: a
Solution: (By law of exponent: (a)^m÷(b)^m = (a÷b)^m

Q.29. For any two non-zero rational numbers x and y, x⁴ ÷ y⁴ is equal to
(a) (x ÷ y)⁰
(b) (x ÷ y)¹
(c) (x ÷ y)⁴
(d) (x ÷ y)⁸
Ans: c
Solution: (By law of exponent: (a)^m÷(b)^m = (a÷b)^m)

Q.30. For a non-zero rational number p, p¹³ ÷ p⁸is equal to
(a) p⁵
(b) p²¹
(c) p^-5
(d) p^-19
Ans: a
Solution: (By law of exponent: (a)^m÷(a)ⁿ = (a)^m-n)

Q.31. For a non-zero rational number z, (z-2)3 equal to
(a) z6
(b) z-6
(c)z1
(d) z4
Ans: b
Solution: (By the law of exponents: (a^m)ⁿ=a^mn)

Q.32. Cube of -1/2 is
(a) 1/8
(b) 1/16
(c) -1/8
(d) -1/16
Ans: c
Solution: Cube of -1/2 = (-1/2)³
= (-1/2) × (-1/2) × (-1/2) = -1/8

Q.33. Which of the following is not the reciprocal of (2/3)⁴?
(a) (3/2)⁴
(b) (3/2)^-4
(c) (2/3)^-4
(d) 3⁴/2⁴
Ans: b
Solution: (2/3)⁴ = 1/(2/3)^-4 = (3/2)^-4

In questions 34 to 50, fill in the blanks to make the statements true.
Q.34. The multiplicative inverse of 10¹⁰ is 10^-10
Q.35. a³ × a^-10 = a^3+(-10) = a^3-10 = a^-7
Q.36. 5⁰ = 1
Q.37. 5⁵ × 5^-5 = 5^5+(-5) = 5^5-5 = 5⁰= 1

Q.38. The value of (1/2³)² equal to (1/2⁶).
Ans: (1/2³)² = (1/2)^3×2 = (1/2)⁶

Q.39. The expression for 8^-2 as a power with the base 2 is (2)^-6
Ans: 8^-2 = (2 × 2 × 2)^-2 = (23)^-2

Q.40. Very small numbers can be expressed in standard form by using negative exponents.
Q.41. Very large numbers can be expressed in standard form by using positive exponents.

42. By multiplying (10)⁵ by (10)^-10 we get 10^-5
Ans: (10)⁵ × (10)^-10 = 10^5+(-10) = 10⁵^-10 = 10^-5

Q.43. [(2/13)^-6÷(2/13)³]³ × (2/13)^-9 = (2/13)^-36
Ans: [(2/13)^-6÷(2/13)³]³ × (2/13)^-9
= [(2/13)^-6^-3]³ × (2/13)^-9
= [(2/13)^-9]³ × (2/13)^-9
= (2/13)^-9×3 × (2/13)^-9
= (2/13)^-27 × (2/13)^-9
= (2/13)^-27-9
= (2/13)^-36

Q.44. Find the value [4^-1 +3^-1 + 6^-2]^-1
Ans: [4^-1 +3^-1 + 6^-2]^-1
= (1/4+1/3+1/6²)^-1
= [(9+12+1)/36]^-1
= (22/36)^-1
= (36/22)

Q.45. [2^-1 + 3^-1 + 4^-1]⁰ = 1 (Using law of exponent, a⁰=1)

Q.46. The standard form of (1/100000000) is 1.0 × 10^-8
Ans: (1/100000000) = 1/1×10⁸ = 1.0 × 10^-8

Q.47. The standard form of 12340000 is 1.234 × 10⁷
Ans: 12340000 = 1234 × 10⁴ = 1.234 × 10³ × 10⁴ = 1.234 × 10⁷

Q.48. The usual form of 3.41 × 10⁶ is 3410000.
Ans: 3.41 × 10⁶ = 3.41 × 10 × 10 × 10 × 10 × 10 × 10
= 341 × 10 × 10 × 10 × 10
= 3410000

Q.49. The usual form of 2.39461 × 10⁶ is 2394610.
Ans: 2.39461 × 10⁶ = 2.39461 × 10 × 10 × 10 × 10 × 10 × 10
= 239461 × 10
= 2394610

Q.50. If 36 = 6 × 6 = 6², then 1/36 expressed as a power with the base 6 is 6^-2.
Ans: 36 = 6 × 6 = 6²
1/36 = 1/6² = 6^-2

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FAQs on NCERT Exemplar Solutions: Exponents & Powers - Mathematics (Maths) Class 8

1. What is the meaning of exponents and powers in mathematics?

Ans. Exponents and powers are mathematical operations that involve raising a number to a certain power or multiplying a number by itself multiple times. In simple terms, an exponent represents how many times a number is multiplied by itself, while a power represents the result of that multiplication.

2. How do you read numbers with exponents and powers?

Ans. Numbers with exponents and powers are read by stating the base number followed by the exponent. For example, "2 to the power of 3" is read as "2 cubed" or "2 raised to the power of 3". Similarly, "10 to the power of 4" is read as "10 to the power of 4" or "10,000".

3. What are the rules of exponents and powers?

Ans. The rules of exponents and powers include: - When multiplying numbers with the same base, add the exponents. - When dividing numbers with the same base, subtract the exponents. - When raising a power to another power, multiply the exponents. - When raising a product to an exponent, distribute the exponent to each factor.

4. How do exponents and powers relate to scientific notation?

Ans. Scientific notation is a way of expressing very large or very small numbers using exponents and powers of 10. In scientific notation, a number is expressed as a product of a decimal number between 1 and 10 and a power of 10. For example, 300,000 can be written as 3 x 10^5 in scientific notation.

5. How are exponents and powers used in real-life applications?

Ans. Exponents and powers are used in various real-life applications such as computing compound interest, calculating population growth, measuring earthquakes on the Richter scale, representing very large or small numbers in scientific fields, and determining the magnitude of electric currents in electronics. They provide a convenient way to represent and manipulate numbers that involve repeated multiplication or division.

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