HP Board Class 9th Number System Questions
Chapter – Number System
Topic – Laws of Exponents for Real Numbers
Question 1. Simplify: 2³ × 2⁴.
Solution:
Using the law of exponents:
aᵐ × aⁿ = aᵐ⁺ⁿ
Therefore,
2³ × 2⁴ = 2³⁺⁴
= 2⁷
= 128
Answer: 128
Question 2. Simplify: 5⁷ ÷ 5³.
Solution:
Using the law of exponents:
aᵐ ÷ aⁿ = aᵐ⁻ⁿ, where a ≠ 0
Therefore,
5⁷ ÷ 5³ = 5⁷⁻³
= 5⁴
= 625
Answer: 625
Question 3. Simplify: (3²)³.
Solution:
Using the law of exponents:
(aᵐ)ⁿ = aᵐⁿ
Therefore,
(3²)³ = 3²ˣ³
= 3⁶
= 729
Answer: 729
Question 4. Simplify: 2⁻³.
Solution:
Using the law of negative exponents:
a⁻ᵐ = 1/aᵐ, where a ≠ 0
Therefore,
2⁻³ = 1/2³
= 1/8
Answer: 1/8
Question 5. Simplify: 4³ × 4⁻¹.
Solution:
Using the law of exponents:
aᵐ × aⁿ = aᵐ⁺ⁿ
Therefore,
4³ × 4⁻¹ = 4³⁺⁽⁻¹⁾
= 4²
= 16
Answer: 16
Question 6. Simplify: (2/3)⁻².
Solution:
Using the law of negative exponents:
(a/b)⁻ᵐ = (b/a)ᵐ, where a and b are non-zero.
Therefore,
(2/3)⁻² = (3/2)²
= 3²/2²
= 9/4
Answer: 9/4
Question 7. Simplify: 2⁵ × 3⁵.
Solution:
Using the law of exponents:
aᵐ × bᵐ = (ab)ᵐ
Therefore,
2⁵ × 3⁵ = (2 × 3)⁵
= 6⁵
= 7776
Answer: 7776
Question 8. Simplify: 7⁰ + 2⁰ + 5⁰.
Solution:
Using the law of zero exponents:
a⁰ = 1, where a ≠ 0
Therefore,
7⁰ + 2⁰ + 5⁰
= 1 + 1 + 1
= 3
Answer: 3
Question 9. Simplify: (2³ × 2⁻⁵) ÷ 2⁻⁴.
Solution:
First, multiply the powers with the same base:
2³ × 2⁻⁵ = 2³⁻⁵
= 2⁻²
Now divide:
2⁻² ÷ 2⁻⁴ = 2⁻²⁻⁽⁻⁴⁾
= 2²
= 4
Answer: 4
Question 10. Simplify: (3² × 3⁻¹)².
Solution:
First, simplify inside the brackets:
3² × 3⁻¹ = 3²⁻¹
= 3¹
Now raise the result to the power of 2:
(3¹)² = 3¹ˣ²
= 3²
= 9
Answer: 9