Moderate Level Worksheet
Part A: Subjective Questions - Very Short Answer (1 Mark Each)
Note: Answer with proper steps and clear explanations. Apply the laws of exponents correctly.
In this moderate level, we'll explore more complex exponent operations and combined applications of multiple laws.
Let's deepen our understanding of exponential operations!
1. Write the product form of 6².
6² =
=
Perfect! 6 squared equals 36.
2. Simplify: 2⁴ × 2³.
2⁴ × 2³ = 2
= 2
Excellent! Add exponents when multiplying with same base.
3. Simplify: (a³ ÷ a²).
a³ ÷ a² = a
= a
=
Correct! a¹ = a.
4. Write the base and exponent in 7⁵.
Base =
Exponent =
Perfect! Identify the base and exponent correctly.
5. What is the value of (10² × 10³)?
10² × 10³ = 10
=
Excellent! 10⁵ = 100,000.
6. Simplify: (5³ ÷ 5²).
5³ ÷ 5² = 5
=
Correct! 5¹ = 5.
7. Write 9³ in expanded form.
9³ =
=
Perfect! 9 × 9 × 9 = 729.
8. Write the reciprocal of 2³.
Reciprocal of 2³ =
=
Excellent! The reciprocal flips the fraction.
9. Express 64 as a power of 4.
64 =
64 =
Correct! 4 × 4 × 4 = 64.
10. Simplify: (3³ × 3²).
3³ × 3² = 3
=
Perfect! 3⁵ = 243.
Drag each operation to its correct law:
Part A: Section B – Short Answer Questions (2 Marks Each)
1. Simplify: (x³ × x² × x).
x³ × x² × x = x
= x
Perfect! Add all exponents: 3 + 2 + 1 = 6.
2. Simplify: (a⁵ ÷ a³).
a⁵ ÷ a³ = a
= a
Excellent! Subtract exponents when dividing.
3. Simplify: (2² × 3³).
Calculate separately (different bases):
2² =
3³ =
4 × 27 =
Great! Different bases: calculate each power first.
4. Simplify: (3²)³.
(3²)³ = 3
= 3
=
Perfect! Power of a power: multiply exponents.
5. Simplify: (5⁰ + 3⁰).
5⁰ =
3⁰ =
1 + 1 =
Excellent! Any nonzero number to power 0 equals 1.
6. Simplify: (10⁴ ÷ 10²).
10⁴ ÷ 10² = 10
=
Correct! 10² = 100.
7. Find the value of (2³ × 5²).
2³ =
5² =
8 × 25 =
Great! Calculate each power, then multiply.
8. Write the laws of exponents for: (a) Multiplication (b) Division
(a) Multiplication: aᵐ × aⁿ =
(b) Division: aᵐ ÷ aⁿ =
Perfect! These are the fundamental laws.
9. Simplify: [(x²)³ ÷ x⁴].
Step 1: (x²)³ = x
Step 2: x⁶ ÷ x⁴ = x
Excellent! Apply power of power first, then division.
10. Simplify: (4³ ÷ 4²) × 4¹.
Step 1: 4³ ÷ 4² = 4
Step 2: 4¹ × 4¹ = 4
=
Perfect! Work step by step.
Part A: Section C – Long Answer Questions (4 Marks Each)
1. Simplify using laws of exponents: (a² × a³) ÷ a².
Step 1: Multiply first
a² × a³ = a
Step 2: Divide
a⁵ ÷ a² = a
Excellent! Final answer: a³.
2. Simplify: [(x³)² × x⁴] ÷ x⁵.
Step 1: (x³)² = x
Step 2: x⁶ × x⁴ = x
Step 3: x¹⁰ ÷ x⁵ = x
Perfect! Multiple operations combined correctly.
3. Simplify: (2³ × 3² × 2²).
Step 1: Combine powers of 2
2³ × 2² = 2
Step 2: Calculate values
2⁵ =
3² =
Step 3: Multiply
32 × 9 =
Excellent! Group same bases, then calculate.
4. Write all laws of exponents with examples.
Law 1: Multiplication
aᵐ × aⁿ =
Example: 2³ × 2² = 2
Law 2: Division
aᵐ ÷ aⁿ =
Example: 5⁴ ÷ 5² = 5
Law 3: Power of Power
(aᵐ)ⁿ =
Example: (3²)³ = 3
Law 4: Power of Product
(a × b)ⁿ =
Example: (2 × 3)² = 2² × 3² =
Perfect! All major laws covered.
5. Simplify: [(4² × 4³) ÷ 4⁴].
Step 1: Multiply
4² × 4³ = 4
Step 2: Divide
4⁵ ÷ 4⁴ = 4
=
Excellent! Brackets first, then divide.
Part B: Objective Questions - Test Your Knowledge!
Answer these multiple choice questions:
6. The value of 2⁰ + 3⁰ =
(a) 0 (b) 2 (c) 1 (d) 3
Correct! 2⁰ = 1 and 3⁰ = 1, so 1 + 1 = 2.
7. (a³ × b³) =
(a) (ab)³ (b) (a + b)³ (c) a³b (d) a²b³
Perfect! Power of a product law: (ab)ⁿ = aⁿ × bⁿ.
8. The reciprocal of 5³ is:
(a) 1/5 (b) 1/25 (c) 1/125 (d) 5
Correct! 5³ = 125, so reciprocal = 1/125.
9. The law of exponents for powers says:
(a) (aᵐ)ⁿ = aᵐ⁺ⁿ (b) (aᵐ)ⁿ = aᵐⁿ (c) (aᵐ × aⁿ) = aᵐⁿ (d) None
Excellent! Power of power: multiply the exponents.
10. 10³ × 10⁴ =
(a) 10⁷ (b) 10¹² (c) 10⁶ (d) 10⁵
Perfect! Add exponents: 3 + 4 = 7.
🌟 Excellent Progress! You've Mastered Intermediate Powers and Exponents!
Here's what you've learned:
Complete Laws of Exponents:
Law 1: Product of Powers (Multiplication)
- aᵐ × aⁿ = aᵐ⁺ⁿ
- Example: x³ × x⁴ = x⁷
- Add exponents when bases are the same
Law 2: Quotient of Powers (Division)
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Example: y⁸ ÷ y³ = y⁵
- Subtract exponents when bases are the same
Law 3: Power of a Power
- (aᵐ)ⁿ = aᵐⁿ
- Example: (2³)² = 2⁶
- Multiply the exponents
Law 4: Power of a Product
- (a × b)ⁿ = aⁿ × bⁿ
- Example: (2 × 3)⁴ = 2⁴ × 3⁴
- Distribute the power to each factor
Law 5: Power of a Quotient
- (a/b)ⁿ = aⁿ/bⁿ
- Example: (3/2)² = 3²/2² = 9/4
- Distribute the power to numerator and denominator
Special Exponent Values:
- a¹ = a (any number to power 1)
- a⁰ = 1 (any nonzero number to power 0)
- 1ⁿ = 1 (1 to any power equals 1)
- 0ⁿ = 0 (0 to any positive power equals 0)
Working with Multiple Operations:
- Apply laws in order: brackets first
- Example: (2³ × 2⁴) ÷ 2⁵
- Step 1: 2³ × 2⁴ = 2⁷
- Step 2: 2⁷ ÷ 2⁵ = 2²
- Step 3: 2² = 4
Different Bases:
- Can't directly combine: calculate each separately
- Example: 2³ × 3² = 8 × 9 = 72
- Find common factors if possible
- Example: 6² = (2 × 3)² = 2² × 3²
Reciprocals and Exponents:
- Reciprocal of aⁿ = 1/aⁿ
- Example: Reciprocal of 2³ = 1/8
- Useful for division: a/b = a × (1/b)
Problem-Solving Strategy:
- Identify which law applies
- Simplify step by step
- Work with same bases when possible
- Calculate final numerical value if needed
- Check if answer can be simplified further
Common Patterns:
- (aⁿ)² = a²ⁿ
- a²ⁿ = (aⁿ)² = (a²)ⁿ
- aⁿ × aⁿ = a²ⁿ = (aⁿ)²
Important Reminders:
- Laws only work with the same base
- Always simplify exponents before calculating
- Keep track of operations: order matters
- Verify answers by expanding if unsure
Mastering these laws makes complex calculations simple and is essential for algebra and beyond!