GCSE Further Maths · Topic 1
Number and algebra
Further Maths at this level stretches your algebra well beyond GCSE.
We will work with surds and indices, factorise cubics with the factor theorem, and master quadratics, inequalities and simultaneous equations.
Then we finish with sequences, algebraic proof and functions.
- 8 subtopics
- 215 questions
- Free: 1.1 notes and 10 questions
In this topic
- 1.1Surds and indices: fractional and negative indices, simplifying and rationalisingFree below · 28 questions
- 1.2Algebraic manipulation: expanding, factorising and algebraic fractionsOpen in Full Marks · 26 questions
- 1.3Polynomials: the factor theorem, the remainder theorem and solving cubicsOpen in Full Marks · 32 questions
- 1.4Quadratics: completing the square, the discriminant and quadratic inequalitiesOpen in Full Marks · 11 questions
- 1.5Simultaneous equations: linear with quadratic, and three linear equationsOpen in Full Marks · 27 questions
- 1.6Sequences: nth terms of linear and quadratic sequences and limiting valuesOpen in Full Marks · 26 questions
- 1.7Algebraic proof: odd, even, consecutive and identity proofsOpen in Full Marks · 29 questions
- 1.8Functions: domain and range, composite and inverse functions, and piecewise functionsOpen in Full Marks · 36 questions
1.1 · Free notes
Surds and indices: fractional and negative indices, simplifying and rationalising
Key points
- Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1.
- Negative indices mean reciprocals: a⁻ⁿ = 1/aⁿ. Fractional indices mean roots: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ. For example 8^(2/3) = (∛8)² = 4.
- A surd is a root that cannot be simplified to a whole number, such as √2. Simplify by taking out square factors: √72 = √36 × √2 = 6√2.
- Surds multiply like this: √a × √b = √(ab), and √a × √a = a. Like surds can be added: 3√5 + 2√5 = 5√5.
- Rationalising the denominator: 1/√a = √a/a; for 1/(a + √b) multiply top and bottom by the conjugate (a − √b), since (a + √b)(a − √b) = a² − b.
- Equations with indices can be solved by writing both sides as powers of the same base, for example 4ˣ = 32 → 2²ˣ = 2⁵ → x = 5/2.
In more detail
- Worked example: simplify (√5 + 1)/(√5 − 1). Multiply top and bottom by (√5 + 1): (5 + 2√5 + 1)/(5 − 1) = (6 + 2√5)/4 = (3 + √5)/2.
- Common mistake: √(a + b) is not √a + √b; for example √(9 + 16) = 5, not 7.
- Exam tip: give surd answers in their simplest form, with a rational denominator, unless the question says otherwise.
- Fractional powers in reverse: to evaluate a^(m/n), take the root first to keep the numbers small, then raise to the power.
Practice
Surds and indices: fractional and negative indices, simplifying and rationalising: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is 27^(2/3)?
- 18
- 6
- 3
- 9
Show answer
D: 9. 27^(1/3) = 3, and 3² = 9.
2. What is 5⁻²?
- −25
- 1/25
- −10
- 25
Show answer
B: 1/25. A negative index means a reciprocal: 1/5² = 1/25.
3. Simplify √50.
- 5√2
- 25√2
- 2√5
- 10√5
Show answer
A: 5√2. √50 = √25 × √2 = 5√2.
4. Simplify √12 + √27.
- √39
- 6√3
- 5√3
- 13√3
Show answer
C: 5√3. √12 = 2√3 and √27 = 3√3, total 5√3.
5. Rationalise 6/√3.
- 6√3
- √3/2
- 18
- 2√3
Show answer
D: 2√3. 6/√3 × √3/√3 = 6√3/3 = 2√3.
6. Expand (3 + √2)(3 − √2).
- 11
- 7
- 9 − √2
- 9 + 2√2
Show answer
B: 7. (a + √b)(a − √b) = a² − b = 9 − 2 = 7.
7. Solve 9ˣ = 27.
- x = 1.5
- x = 3
- x = 2
- x = 3/4
Show answer
A: x = 1.5. 9ˣ = 3²ˣ and 27 = 3³, so 2x = 3 and x = 1.5.
8. What is 16^(−1/2)?
- −4
- 4
- 1/4
- −1/4
Show answer
C: 1/4. 16^(1/2) = 4, so 16^(−1/2) = 1/4.
9. Simplify (x³)⁴ ÷ x⁵.
- x²
- x¹²
- x⁶⁰
- x⁷
Show answer
D: x⁷. (x³)⁴ = x¹², and x¹² ÷ x⁵ = x⁷.
10. Rationalise 1/(2 + √3).
- 2 + √3
- 2 − √3
- (2 − √3)/7
- 1/7
Show answer
B: 2 − √3. Multiply by (2 − √3): denominator 4 − 3 = 1, so the answer is 2 − √3.
Keep going in Full Marks
Full Marks has 28 questions on Surds and indices: fractional and negative indices, simplifying and rationalising (18 more than here), set out in practice levels. You also get notes and 3D models for all 8 subtopics of Number and algebra, plus games, timed exams and past papers.