Indices II (Copy)
IGCSE Mathematics 0580 – Indices Practice Exam
Divided into Paper 1 (Non-Calculator) and Paper 2 (Calculator Allowed)
Full Worked Solutions Provided
PAPER 1 – NON-CALCULATOR SECTION
Simplify Expressions
Q1. Simplify:
7² × 7³
Solution:
Use the rule aᵐ × aⁿ = aᵐ⁺ⁿ
7² × 7³ = 7²⁺³ = 7⁵ = 16807
Answer: 16807
Q2. Simplify:
4⁵ ÷ 4²
Solution:
Use the rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ
4⁵ ÷ 4² = 4⁵⁻² = 4³ = 64
Answer: 64
Q3. Simplify:
(3x²)³
Solution:
Apply power to both terms:
= 3³ × (x²)³
= 27x⁶
Answer: 27x⁶
Q4. Simplify:
a⁻³
Solution:
Negative power means reciprocal:
= 1 ÷ a³
Answer: 1/a³
Q5. Find the value of x:
2ˣ = 32
Solution:
Recognize that 32 = 2⁵
So 2ˣ = 2⁵
Thus, x = 5
Answer: 5
PAPER 2 – CALCULATOR ALLOWED SECTION
Simplify and Solve
Q6. Simplify:
(5a³)²
Solution:
Apply the power:
= 5² × (a³)²
= 25a⁶
Answer: 25a⁶
Q7. Simplify:
12a⁵ ÷ 3a⁻²
Solution:
Simplify numbers: 12 ÷ 3 = 4
Apply index rule: a⁵ ÷ a⁻² = a⁵⁻(–2) = a⁷
Thus: 4a⁷
Answer: 4a⁷
Q8. Simplify:
6x⁷y⁴ × 5x⁻⁵y
Solution:
Multiply numbers: 6 × 5 = 30
Apply indices separately:
x⁷ × x⁻⁵ = x⁷⁻⁵ = x²
y⁴ × y¹ = y⁵
Thus: 30x²y⁵
Answer: 30x²y⁵
Q9. Simplify:
(2x⁴)³
Solution:
= 2³ × (x⁴)³
= 8x¹²
Answer: 8x¹²
Q10. Simplify:
(3a²b³)²
Solution:
= 3² × (a²)² × (b³)²
= 9a⁴b⁶
Answer: 9a⁴b⁶
Q11. Simplify:
15x⁵ ÷ 3x²
Solution:
15 ÷ 3 = 5
x⁵ ÷ x² = x³
Thus: 5x³
Answer: 5x³
Q12. Solve for x:
3ˣ = 81
Solution:
81 = 3⁴
Thus, 3ˣ = 3⁴
So, x = 4
Answer: 4
Q13. Simplify:
(4a⁻²b³)²
Solution:
Apply power:
4² × (a⁻²)² × (b³)²
= 16 × a⁻⁴ × b⁶
Answer: 16a⁻⁴b⁶
Q14. Simplify:
(2x³y²)² ÷ (4x²y⁵)
Solution:
First expand numerator:
(2)² × (x³)² × (y²)² = 4x⁶y⁴
Thus: (4x⁶y⁴) ÷ (4x²y⁵)
Simplify numbers: 4 ÷ 4 = 1
Apply indices:
x⁶ ÷ x² = x⁴
y⁴ ÷ y⁵ = y⁻¹
Thus: x⁴/y
Answer: x⁴/y
Q15. Simplify:
(5m⁻²n³)⁻¹
Solution:
Negative exponent: reciprocal
= 1 ÷ (5m⁻²n³)
= (1/5) × m² ÷ n³
Thus: m² / (5n³)
Answer: m² / (5n³)
This covers all key types of exam questions for indices at Core and Extended level without using logarithms.