Indices and surds

The method

1. Add the powers when multiplying and subtract them when dividing.
2. Multiply the powers when a power is raised to another power.
3. Turn a negative power into a reciprocal: a⁻ⁿ = 1/aⁿ, and remember a⁰ = 1.
4. Read a fractional power as a root and a power, so a^1/n is the nth root of a.
5. Take the largest square factor out of a surd, and multiply top and bottom by the conjugate to rationalise a denominator.

Mistakes that lose marks

• Writing 2³ × 2⁴ as 2¹²: the powers are multiplied only when one power is raised to another power.
• Leaving √50 as the answer when the question asks for the simplest form, instead of 5√2.
• Writing 3⁻² as −9. A negative power gives a reciprocal, so the answer is 1/9.
• Rationalising a denominator such as 3/(2 + √5) by multiplying by √5 instead of by the conjugate 2 − √5.
• Forgetting that a⁰ = 1, and treating the zero power as though the answer were 0.

7 exam-style questions

These are our own questions written in the style of the exam, not past-paper questions. The mark codes are M = method, A = accuracy and B = independent mark. Try each one, then open the solution.

1. Exam-style core Evaluate (a) 6⁸ ÷ 6⁵ (b) (3²)⁴ [3 marks]
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   1. (a) Subtract the powers: 6⁸ ÷ 6⁵ = 6^{8 − 5} = 6³ M1
   2. (b) Multiply the powers: (3²)⁴ = 3^{2 × 4} = 3⁸ M1
   3. 6³ = 216 and 3⁸ = 6561 A1

   Answer: (a) 216 (b) 6561

2. Exam-style extended Evaluate (a) 7⁰ (b) 5⁻² (c) (2/5)⁻¹ [4 marks]
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   1. (a) Any non-zero number to the power 0 is 1, so 7⁰ = 1 A1
   2. (b) A negative power gives a reciprocal: 5⁻² = 1/5² M1
   3. (b) 1/5² = 1/25 A1
   4. (c) Invert the fraction: (2/5)⁻¹ = 5/2 A1

   Answer: (a) 1 (b) 1/25 (c) 5/2

3. Exam-style extended Evaluate (a) 49^1/2 (b) 27^1/3 (c) 16^3/4 [3 marks]
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   1. (a) A power of 1/2 is a square root: 49^1/2 = √49 = 7 B1
   2. (b) A power of 1/3 is a cube root: 27^1/3 = 3 B1
   3. (c) 16^3/4 = (16^1/4)³ = 2³ = 8 A1

   Answer: (a) 7 (b) 3 (c) 8

4. Exam-style extended Simplify √50 + √18 − √8, giving your answer in the form k√2. [3 marks]
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   1. √50 = √(25 × 2) = 5√2 and √18 = √(9 × 2) = 3√2 M1
   2. √8 = √(4 × 2) = 2√2, so the expression is 5√2 + 3√2 − 2√2 M1
   3. 5√2 + 3√2 − 2√2 = 6√2 A1

   Answer: 6√2

5. Exam-style extended Rationalise the denominator of (a) 1/√3 (b) 10/√5 [4 marks]
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   1. (a) Multiply top and bottom by √3: 1/√3 × √3/√3 = √3/3 M1
   2. (a) = √3/3 A1
   3. (b) Multiply top and bottom by √5: 10/√5 × √5/√5 = 10√5/5 M1
   4. (b) 10√5/5 = 2√5 A1

   Answer: (a) √3/3 (b) 2√5

6. Exam-style extended (a) Expand and simplify (2 + √3)². (b) Rationalise the denominator of 3/(2 + √5). [5 marks]
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   1. (a) (2 + √3)² = 4 + 4√3 + 3 M1
   2. (a) = 7 + 4√3 A1
   3. (b) Multiply top and bottom by the conjugate 2 − √5: 3(2 − √5)/((2)² − (√5)²) M1
   4. (b) The denominator is 4 − 5 = −1, so the fraction is −3(2 − √5) M1
   5. (b) = 3√5 − 6 A1

   Answer: (a) 7 + 4√3 (b) 3√5 − 6

7. Exam-style higher (a) Given that 8^2/3 = 2^k, find the value of k. (b) Rationalise the denominator of (√5 + 1)/(√5 − 1), giving your answer in the form (a + b√5)/c. [5 marks]
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   1. (a) 8^2/3 = (8^1/3)² = 2² M1
   2. (a) So k = 2 A1
   3. (b) Multiply top and bottom by √5 + 1: (√5 + 1)²/((√5)² − 1²) M1
   4. (b) = (5 + 2√5 + 1)/4 = (6 + 2√5)/4 M1
   5. (b) Divide top and bottom by 2: (3 + √5)/2 A1

   Answer: (a) k = 2 (b) (3 + √5)/2

Where this is in the syllabus

Indices and surds appears in Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1. For the official specifications and past papers, use the board's own website: Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1.

Questions students ask