IGCSE

Algebra and Graphs

Exercise: 11.Algebraic representation and manipulation & 12 Algebraic indices

IGCSE Mathematics (Core) – Algebra Practice

Questions 1–30

1. (a) Simplify (2x³y^-2)^-3.
   (b) Evaluate 2³ × 2^-5.
2. Expand and simplify 5x(3x – 2) – (2x – 3)(x + 4).
3. Factorise completely 4x² – 36.
4. Simplify (3x² + 15x) / (x² + 2x – 15).
5. Make x the subject of the formula: y = (2x + 3) / 5.
6. Simplify fully: (3x – 2) / (x + 4) – (2x + 1) / (x + 4).
7. Solve the equation: x² – 7x + 12 = 0.
8. Solve for x: 4^x × 8^x-1 = 2^3x+2.
9. Expand and simplify: (3x – 4)² + (2x + 5)(2x – 5).
10. Simplify (2x³y)² / (4x⁵y³).
11. Given f(x) = x² – 4, find and simplify f(2x + 1).
12. A cube has a side length of (2x – 1) cm.
    (a) Write down an expression for the volume of the cube. (b) Given that the volume is 27 cm³, find the value of x.
13. Write as a single power of x:
    (a) x³ × x^-2 (b) x⁴ ÷ x^-3 (c) (x²)⁵
14. Simplify fully: (x² – 16) / (x² + 8x + 16).
15. Evaluate (2/3)^-3.

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16. Simplify (3a²b)³ × 2a^-4b².
17. Expand and simplify (2x + 3)(x – 5) – x(x – 2).
18. Factorise completely 2x² – 7x – 15.
19. Simplify (x² + 3x – 10) / (x² – 25).
20. Make L the subject of the formula: T = 2π √(L / g).
21. Simplify fully: [3 / (x+2)] × [4 / (x-5)].
22. Solve the equation: x² – 5x – 14 = 0.
23. Solve for x: 9^x = 27^x-2.
24. Expand and simplify: (2x – 5)(3x + 4) – (x – 1)².
25. Factorise completely 3x² – 48.
26. Simplify (3x²y³)³ / (27x⁵y⁷).
27. Simplify (x³ × x^-5) / x^-2.
28. Given f(x) = x² – 4x, find and simplify f(2x + 1).
29. Write (x²)⁵ as a single power of x.
30. Evaluate (3/5)^-2.