Before going into solving Exercise 10.1 Exponents and Powers, lets have a look at some of the basic concepts.
Powers with Negative Exponents:
Definition: When a number is raised to a negative exponent, it’s the reciprocal of the base raised to the absolute value of the exponent.
Formula: a^(-n) = 1 / (aⁿ) where a is the base and n is a positive integer.
Example: 2^(-3) = 1 / (2³) = 1 / 8.
Laws of Exponents:
Product of Powers (Same Base): Add exponents when multiplying powers with the same base.
Formula: aᵐ * aⁿ = a^(m+n).
Example: 3² * 3³ = 3^(2+3) = 3⁵ = 243.
Quotient of Powers (Same Base): Subtract the exponent of the denominator from the exponent of the numerator when dividing powers with the same base.
Formula: aᵐ / aⁿ = a^(m-n).
Example: 2⁵ / 2² = 2^(5-2) = 2³ = 8.
Power of a Power: Multiply the exponents when raising a power to a power.
Formula: (aᵐ)ⁿ = a^(mn).
Example: (4²)³ = 4^(2*3) = 4⁶ = 4096.
Power of a Product: Raise each factor to the power when raising a product to a power.
Formula: (ab)ⁿ = aⁿ * bⁿ.
Example: (2 * 3)² = 2² * 3² = 4 * 9 = 36.
Power of a Quotient: Raise both the numerator and the denominator to the power when raising a quotient to a power.
Formula: (a/b)ⁿ = aⁿ / bⁿ.
Example: (3/2)² = 3² / 2² = 9 / 4.
NCERT Solutions for Class 8 Maths Exercise 10.1 Chapter 10 Exponents and Powers
1. Evaluate:
(i) 3⁻²
(ii) (-4)⁻²
(iii) (1/2)⁻⁵
(i) 3⁻²
For a negative exponent, take the reciprocal of the base raised to the positive of that exponent.
Solution:
3⁻²
= 1/(3²) [a⁻ᵐ = 1/aᵐ]
= 1/9.
(ii) (-4)⁻²
Take the reciprocal of the base raised to that exponent.
Solution:
(-4)⁻²
= 1/((-4)²) [a⁻ᵐ = 1/aᵐ]
= 1/16.
(iii) (1/2)⁻⁵
A negative exponent on a fraction inverts the fraction and converts the exponent to positive.
Solution:
(1/2)⁻⁵
= 1⁻⁵/2⁻⁵ [(a/b)⁻ᵐ = a⁻ᵐ/b⁻ᵐ]
= 2⁵/1⁵ [a⁻ᵐ = 1/aᵐ]
= 2⁵
= 32.
2. Simplify and express the result in power notation with positive exponent:
(i) (-4)⁻⁵ ÷ (-4)⁸
(ii) (1/2³)²
(iii) (-3)⁴ × (5/3)⁴
(iv) (3⁻⁷ ÷ 3⁻¹⁰) × 3⁻⁵
(v) 2⁻³ × (-7)⁻³
(i) (-4)⁵ ÷ (-4)⁸
When dividing with the same base, subtract the exponents.
Solution:
(-4)⁵/(-4)⁸
= 1/((-4)⁻⁵ × (-4)⁸) [a⁻ᵐ = 1/aᵐ]
= 1/(-4)⁸⁻⁵ [aᵐ × aⁿ = aᵐ⁺ⁿ]
= 1/(-4)³
(ii) (1/2³)²
When raising a power to a power, multiply the exponents.
Solution:
(1/2³)²
= 1²/(2³)² [(a/b)ᵐ = aᵐ/bᵐ]
= 1/2⁶
(iii) (-3)⁴ × (5/3)⁴
Solution:
(-3)⁴ × (5/3)⁴
= (-1 × 3)⁴ × (5/3)⁴ [-a = -1 × a]
= (-1)⁴ × 3⁴ × 5⁴/3⁴ [(a/b)ᵐ = aᵐ/bᵐ, (a × b)ᵐ = aᵐ × bᵐ]
= 1 × 5⁴
(iv) (3⁻⁷ ÷ 3⁻¹⁰) × 3⁻⁵
Solution:
(3⁻⁷ ÷ 3⁻¹⁰) × 3⁻⁵
= 3⁻⁷ × 3¹⁰ × 3⁻⁵ [a⁻ᵐ ÷ a⁻ⁿ = a⁻ᵐ × aⁿ]
= 3⁻⁷+¹⁰⁻⁵ [When multiplying powers with the same base, add their exponents]
= 3⁻² [Simplifying the exponent: -7 + 10 – 5]
= 1/3² [a⁻ᵐ = 1/aᵐ]
(v) 2⁻³ × (-7)⁻³
When multiplying with negative exponents, take the reciprocal of each base raised to the positive exponent and then multiply.
Solution:
2⁻³ × (-7))⁻³
= (2 × (-7))⁻³ [aᵐ × bᵐ = (a × b)ᵐ]
= (-14)⁻³
= 1/-14³ [a⁻ᵐ = 1/aᵐ]
3. Find the value of:
(i) (3⁰ + 4⁻¹) × 2²
(ii) (2⁻¹ × 4⁻¹) ÷ 2⁻²
(iii) 1/2⁻² + 1/3⁻² + 1/4⁻²
(i) (3⁰ + 4⁻¹) × 2²
Apply the exponent rules, remembering that any number to the power of 0 is 1, and then perform the addition and multiplication.
Solution:
(3⁰ + 4⁻¹) × 2²
= (1 + 1/4) × 4 [a⁰ = 1 and a⁻ᵐ = 1/aᵐ]
= 5/4 × 4 = 5.
(ii) (2⁻¹ × 4⁻¹) ÷ 2⁻²
For multiplication, add the exponents. For division, subtract the exponents.
Solution:
(2⁻¹ × 4⁻¹) ÷ 2⁻²
= 1/2 × 1/4 ÷ 1/2² [a⁻ᵐ = 1/aᵐ]
= 1/8 ÷ 1/4
= (1/8) × 4
= 1/4
(iii) 1/2⁻² + 1/3⁻² + 1/4⁻²
For a negative exponent in the denominator, take the reciprocal to make it a positive exponent.
Solution:
1/2⁻² + 1/3⁻² + 1/4⁻²
= 2² + 3² + 4² [1/a⁻ᵐ = aᵐ]
= 4 + 9 + 16
= 29.
(iv) (3⁻¹ + 4⁻¹ + 5⁻¹⁰)
Calculate the reciprocal of each number and then add them.
Solution:
1/3 + 1/4 + 1/5¹⁰ [a⁻ᵐ = 1/aᵐ]
= 1/3 + 1/4 + 1/9765625
= 0.5833 (approx)
(v) {(-2/3)⁻²}²
Solution:
(3/-2 × 3/-2)²
= (9/4)²
= 81/16
4. Evaluate:
(i) (8⁻¹ × 5³) ÷ 2⁻⁴
Apply the reciprocal to 8⁻¹, multiply by 5 cubed, and divide by the reciprocal of 2⁻⁴.
Solution:
(8⁻¹ × 5³) ÷ 2⁻⁴
Rewrite each term in its reciprocal form where necessary:
8⁻¹ = 1/8 [a⁻ᵐ = 1/aᵐ]
2⁻⁴ = 1/2⁴ [a⁻ᵐ = 1/aᵐ]
Combine the terms:
(1/8 × 5³) ÷ 1/2⁴
= (1/8 × 125) ÷ 1/16
= 125/8 ÷ 1/16
Perform division by multiplying with the reciprocal:
125/8 × 16/1
= 125 × 2
Perform the multiplication:
125 × 2
= 250
(ii) (5⁻¹ × 2⁻¹) × 6⁻¹
Multiply the reciprocals of 5, 2, and 6.
Solution:
(5⁻¹ × 2⁻¹) × 6⁻¹
Rewrite each term in its reciprocal form:
5⁻¹ = 1/5 [a⁻ᵐ = 1/aᵐ]
2⁻¹ = 1/2 [a⁻ᵐ = 1/aᵐ]
6⁻¹ = 1/6 [a⁻ᵐ = 1/aᵐ]
Combine the terms:
(1/5 × 1/2) × 1/6
= 1/(5 × 2 × 6)
Perform the multiplication:
1/(5 × 2 × 6)
= 1/60
5. Find the value of m for which 5ᵐ ÷ 5⁻³ = 5⁵.
Apply the laws of exponents for division and set the expression equal to 5⁵.
Solution:
To find the value of m for which 5ᵐ ÷ 5⁻³ = 5⁵:
Rewrite the equation using the property of exponents:
5ᵐ ÷ 5⁻³ = 5⁵
5ᵐ ÷ 1/5³ = 5⁵ [a⁻ᵐ = 1/aᵐ]
5ᵐ × 5³ = 5⁵ (since dividing by a number is the same as multiplying by its reciprocal)
Combine the terms using the exponent rule (aⁿ × aᵐ = aⁿ⁺ᵐ):
5ᵐ × 5³ = 5⁵
5ᵐ⁺³ = 5⁵
Since the bases are the same, set the exponents equal to each other:
m + 3 = 5
Solve for m:
m = 5 – 3
m = 2
6. Evaluate:
(i) {(1/3)⁻¹ – (1/4)⁻¹}⁻¹
Step 1: Calculate the negative reciprocal of each fraction.
(1/3)⁻¹ = 3
(1/4)⁻¹ = 4
Step 2: Subtract the second result from the first.
3 – 4 = -1
Step 3: Find the negative reciprocal of the result.
(-1)⁻¹ = 1/(-1)¹ = -1 [a⁻ᵐ = 1/aᵐ]
Answer: -1
(ii) (5/8)⁻⁷ × (8/5)⁻⁴
Raise each fraction to the negative power and then multiply.
Solution:
(5/8)⁻⁷ × (8/5)⁻⁴
Simplify each term:
(5/8)⁻⁷ = (8/5)⁷
(8/5)⁻⁴ = (5/8)⁴
Combine terms:
(8/5)⁷ × (5/8)⁴
Simplify further:
= (8⁷ / 5⁷) × (5⁴ / 8⁴)
= 8⁷ × 5⁴ / (5⁷ × 8⁴)
= 8³ / 5³
= 512 / 125
7. Simplify:
(i) (25 × t⁻⁴) ÷ (5⁻³ × 10 × t⁻⁸)
Simplify the expression by canceling out common factors and applying the laws of exponents.
Solution:
Expression (25 × t⁻⁴) ÷ (5⁻³ × 10 × t⁻⁸) simplifies to:
Simplify each term:
5⁻³ = 1/5³
Combine terms:
(25 × t⁻⁴) ÷ (1/5³ × 10 × t⁻⁸)
= 25 × t⁻⁴ × 5³ ÷ (10 × t⁻⁸)
Simplify further:
= (25 × 5³ × t⁻⁴) ÷ (10 × t⁻⁸)
= (25 × 125 × t⁻⁴) ÷ (10 × t⁻⁸)
Final result:
= (3125 × t⁻⁴) ÷ (10 × t⁻⁸)
= 312.5t⁴
(ii) 3⁻⁵ × 10⁻⁵ × 125 ÷ (5⁻⁷ × 6⁻⁵)
Multiply the negative exponents and then divide by the reciprocal of the negative exponents.
Solution:
The expression 3⁻⁵ × 10⁻⁵ × 125 ÷ (5⁻⁷ × 6⁻⁵) simplifies to:
Simplify each term:
3⁻⁵ = 1/3⁵
10⁻⁵ = 1/10⁵
5⁻⁷ = 1/5⁷
6⁻⁵ = 1/6⁵
Combine terms:
1/3⁵ × 1/10⁵ × 125 ÷ (1/5⁷ × 1/6⁵)
Simplify further:
(125 / (3⁵ × 10⁵)) × (5⁷ × 6⁵)
Final result: 3125
Practice Worksheet with Challenging Questions For Class 8 Maths for Exercise 10.1 Exponents and Powers
Questions
- Evaluate: (i) (-5)⁻³ (ii) (2/3)⁻⁴ (iii) (-1/2)⁻⁶
- Simplify and express with positive exponent: (i) (-3)⁻⁷ ÷ (-3)⁹ (ii) (2⁻⁴)³ (iii) (-2)⁵ × (3/2)⁵ (iv) (4⁻⁹ ÷ 4⁻¹²) × 4⁻³ (v) 3⁻⁴ × (-6)⁻⁴
- Find the value of: (i) (4⁰ + 5⁻¹) × 3² (ii) (3⁻¹ × 6⁻¹) ÷ 3⁻² (iii) 1/4⁻³ + 1/5⁻³ + 1/6⁻³
- Evaluate: (i) (9⁻¹ × 6³) ÷ 3⁻⁵ (ii) (6⁻¹ × 3⁻¹) × 7⁻¹
- Find the value of n for which 7ⁿ ÷ 7⁻² = 7⁶.
- Evaluate: (i) {(1/5)⁻¹ – (1/6)⁻¹}⁻¹ (ii) (7/9)⁻⁶ × (9/7)⁻³
- Simplify: (i) (36 × s⁻⁵) ÷ (6⁻⁴ × 12 × s⁻⁹) (ii) 4⁻⁶ × 15⁻⁶ × 216 ÷ (6⁻⁸ × 9⁻⁶)
- Evaluate: (i) 8⁻³ (ii) (-3/4)⁻⁵ (iii) (-2/5)⁻⁷
- Simplify: (i) (-2)⁻⁴ ÷ (-2)⁷ (ii) (5⁻⁵)² (iii) (3)⁻² × (4/3)⁻²
- Find the value of: (i) (7⁰ + 8⁻¹) × 4² (ii) (4⁻¹ × 8⁻¹) ÷ 4⁻³ (iii) 1/7⁻³ + 1/8⁻³ + 1/9⁻³
- Evaluate: (i) (16⁻¹ × 7³) ÷ 4⁻² (ii) (7⁻¹ × 3⁻¹) × 8⁻¹
- Find the value of m for which 8ᵐ ÷ 8⁻⁴ = 8⁷.
- Evaluate: (i) {(1/7)⁻¹ – (1/8)⁻¹}⁻¹ (ii) (6/11)⁻⁸ × (11/6)⁻⁵
- Simplify: (i) (49 × u⁻⁶) ÷ (7⁻⁵ × 14 × u⁻¹¹) (ii) 5⁻⁷ × 20⁻⁷ × 3125 ÷ (25⁻⁹ × 10⁻⁷)
- Evaluate: (i) 27⁻² (ii) (-4/5)⁻⁶ (iii) (-3/7)⁻⁴
Answers
- Evaluate: (i) (-5)⁻³ (ii) (2/3)⁻⁴ (iii) (-1/2)⁻⁶
- Simplify and express with positive exponent: (i) (-3)⁻⁷ ÷ (-3)⁹ (ii) (2⁻⁴)³ (iii) (-2)⁵ × (3/2)⁵ (iv) (4⁻⁹ ÷ 4⁻¹²) × 4⁻³ (v) 3⁻⁴ × (-6)⁻⁴
- Find the value of: (i) (4⁰ + 5⁻¹) × 3² (ii) (3⁻¹ × 6⁻¹) ÷ 3⁻² (iii) 1/4⁻³ + 1/5⁻³ + 1/6⁻³
- Evaluate: (i) (9⁻¹ × 6³) ÷ 3⁻⁵ (ii) (6⁻¹ × 3⁻¹) × 7⁻¹
- Find the value of n for which 7ⁿ ÷ 7⁻² = 7⁶.
- Evaluate: (i) {(1/5)⁻¹ – (1/6)⁻¹}⁻¹ (ii) (7/9)⁻⁶ × (9/7)⁻³
- Simplify: (i) (36 × s⁻⁵) ÷ (6⁻⁴ × 12 × s⁻⁹) (ii) 4⁻⁶ × 15⁻⁶ × 216 ÷ (6⁻⁸ × 9⁻⁶)
- Evaluate: (i) 8⁻³ (ii) (-3/4)⁻⁵ (iii) (-2/5)⁻⁷
- Simplify: (i) (-2)⁻⁴ ÷ (-2)⁷ (ii) (5⁻⁵)² (iii) (3)⁻² × (4/3)⁻²
- Find the value of: (i) (7⁰ + 8⁻¹) × 4² (ii) (4⁻¹ × 8⁻¹) ÷ 4⁻³ (iii) 1/7⁻³ + 1/8⁻³ + 1/9⁻³
- Evaluate: (i) (16⁻¹ × 7³) ÷ 4⁻² (ii) (7⁻¹ × 3⁻¹) × 8⁻¹
- Find the value of m for which 8ᵐ ÷ 8⁻⁴ = 8⁷.
- Evaluate: (i) {(1/7)⁻¹ – (1/8)⁻¹}⁻¹ (ii) (6/11)⁻⁸ × (11/6)⁻⁵
- Simplify: (i) (49 × u⁻⁶) ÷ (7⁻⁵ × 14 × u⁻¹¹) (ii) 5⁻⁷ × 20⁻⁷ × 3125 ÷ (25⁻⁹ × 10⁻⁷)
- Evaluate: (i) 27⁻² (ii) (-4/5)⁻⁶ (iii) (-3/7)⁻⁴
