AQA A-level Mathematics: Rational Indices and Surds (7357) – Revision Notes

These revision notes condense content references B1 (laws of indices for all rational exponents) and B2 (using and manipulating surds, including rationalising the denominator) from section 3.3 B: Algebra and functions of the AQA A-level Mathematics (7357) specification (version 1.3, dated 31 January 2018), which applies to A-level exams from June 2018. 7357 is a linear A-level with no AS/A split. Section B belongs to the Paper 1 content, and Papers 2 and 3 can assess any Paper 1 content.

For full explanations and worked examples, read the Rational indices and surds study guide. When you are ready to test yourself, use the practice questions. Course links: AQA A-level Mathematics hub, printable checklist and free 10-minute diagnostics.

Coverage at a glance

Reference	Skill	Section below
B1	Laws of indices, including negative, zero and fractional exponents	Index laws
B1	Simplifying powers of numbers and letters	Fractional indices
B1	Solving index equations	Solving index equations
B2	Simplifying and combining surds	Surd rules
B2	Rationalising k/√a and k/(a + √b)	Rationalising

Index laws

Law	Note
aᵐ × aⁿ = aᵐ⁺ⁿ	Same base only
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Same base only
(aᵐ)ⁿ = aᵐⁿ	Multiply the indices
a⁰ = 1	For a ≠ 0
a⁻ⁿ = 1/aⁿ	Reciprocal, not negative
(ab)ⁿ = aⁿbⁿ	Apply to every factor, including the number
(a/b)ⁿ = aⁿ/bⁿ	Apply to top and bottom

The first three rows are the core laws, and the rest follow from them. Learn all of them.

Fractional indices

a^(1/n) = ⁿ√a
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
a^(−m/n) = 1 / a^(m/n)

Method in steps

1. Deal with the sign of the index first: a negative index means turn the fraction over.
2. Take the root of the base.
3. Apply the power.
4. Check with a quick estimate. A number to a positive power above 1 gets bigger when a > 1.

Reminder: 64^(2/3) = (³√64)² = 4² = 16. Taking the root before the power keeps the numbers small.

Reminder: with letters, apply the index to each factor. (8a³)^(2/3) = 4a².

Solving index equations

Type	What to do
Unknown in the index, common base available	Rewrite each side with that base, then set the indices equal
Unknown as the base with a fractional index	Raise both sides to the reciprocal power
Powers of 2ˣ, 3ˣ, 5ˣ in a quadratic pattern	Substitute y = aˣ, solve for y, convert back, reject y ≤ 0

Reminder (same base): 4ˣ = 1/32 means 2^(2x) = 2⁻⁵, so 2x = −5 and x = −5/2.

Reminder (unknown as base): x^(−1/2) = 1/7 gives x = (1/7)⁻² = 49, with x > 0.

Reminder (quadratic pattern): 2^(2x) − 10 × 2ˣ + 16 = 0. Let y = 2ˣ: y² − 10y + 16 = 0, so (y − 2)(y − 8) = 0. Then 2ˣ = 2 or 2ˣ = 8, so x = 1 or x = 3.

Surd rules

Rule	Example
√(ab) = √a × √b	√45 = √9 × √5 = 3√5
√(a/b) = √a / √b	√(3/16) = √3 / 4
(√a)² = a	(√6)² = 6
√a × √a = a	√7 × √7 = 7
√(a + b) ≠ √a + √b	√(9 + 16) = 5 but √9 + √16 = 7

Method in steps: simplifying and collecting

1. Write the number under the root as (a square number) × (the rest).
2. Take the square root of the square factor outside.
3. Collect terms with the same surd as like terms.

Reminder: √12 + √27 = 2√3 + 3√3 = 5√3.

Method in steps: expanding

Multiply out as with algebra, then replace each √a × √a by a, and collect the rational part and the surd part.

Reminder: (3 − √2)² = 9 − 6√2 + 2 = 11 − 6√2.

Rationalising

Denominator	Multiply top and bottom by	Result of the multiplication below
√a	√a	a
a + √b	a − √b	a² − b
a − √b	a + √b	a² − b
√a + √b	√a − √b	a − b

Method in steps

1. Simplify the denominator if you can, for example √12 = 2√3.
2. Identify the conjugate (change the sign between the two terms).
3. Multiply numerator and denominator by it.
4. Expand the numerator carefully. Use the difference of two squares on the denominator.
5. Divide through by any common factor and write the answer in the form a + b√c.

Reminder: 10/(3 + √2) = 10(3 − √2)/(9 − 2) = (30 − 10√2)/7.

Reminder: 2/(√5 + √3) = 2(√5 − √3)/(5 − 3) = √5 − √3.

Reminder: (1 + √3)/(√3 − 1). Multiply by (√3 + 1)/(√3 + 1). Numerator: (1 + √3)² = 4 + 2√3. Denominator: 3 − 1 = 2. Result: 2 + √3.

Longer worked reminders

Letters and fractional powers together

Simplify (27a⁶b⁻³)^(−2/3) × 6a⁴b.

27^(−2/3) = 1/(³√27)² = 1/9
(a⁶)^(−2/3) = a⁻⁴
(b⁻³)^(−2/3) = b²

The bracket is (1/9)a⁻⁴b². Multiply by 6a⁴b:

(6/9) × a⁻⁴⁺⁴ × b²⁺¹ = (2/3)b³

The powers of a cancel completely, because a⁻⁴ × a⁴ = a⁰ = 1. Answer: 2b³/3. Check that you applied the index to the number 27 as well as to the letters.

Rationalising with a surd in the numerator

Write (2√3 + 1)/(√3 − 1) in the form p + q√3.

The conjugate of √3 − 1 is √3 + 1.

Numerator:   (2√3 + 1)(√3 + 1) = 6 + 2√3 + √3 + 1 = 7 + 3√3
Denominator: (√3 − 1)(√3 + 1) = 3 − 1 = 2

Answer: (7 + 3√3)/2, so p = 7/2 and q = 3/2. Fractions are allowed for p and q. Do not force the answer into integers.

Using conjugates to add fractions

1/(2 − √3) + 1/(2 + √3) = (2 + √3) + (2 − √3), all over (4 − 3), which is 4. Notice that the surds cancel. Adding a number and its conjugate always gives a rational answer, which is a quick way to spot an error.

Links to other topics

Index laws appear again in exponentials and logarithms, where you rewrite a^x in different bases. Surds appear in the quadratic formula, in exact trigonometric values and in distance calculations. The exponentials and logarithms notes build directly on B1.

Must-know distinctions

• Same base is needed to add or subtract indices. 2³ × 5³ = 10³, but 2³ × 5² has no single-power form.
• Negative index gives a reciprocal. A negative base gives a negative number (or a positive one, depending on the power). They are different things.
• Surd versus rational: √49 = 7 is rational, √48 = 4√3 is a surd.
• Simplified versus rationalised: 5/√2 is correct but not rationalised. 5√2/2 is the form to give.
• Exact versus decimal: if the question says “exact” or “in the form a + b√c”, give the surd form, because a rounded decimal does not answer the question.

Quick self-test

Do these without a calculator. Cover the answers first.

1. Evaluate 64^(−5/6).
2. Evaluate (9/16)^(3/2).
3. Write √45 in the form k√5.
4. Expand and simplify (3 − √2)².
5. Rationalise 10/(3 + √2).
6. Solve 4ˣ = 1/32.
7. Solve x^(−1/2) = 1/7, where x > 0.
8. Evaluate 8^(2/3) × 2⁻³.
9. Simplify √12 + √27.
10. Write (1 + √3)/(√3 − 1) in the form a + b√3.
11. Solve 25ˣ = 5^(x+3).
12. Solve 2^(2x) − 10 × 2ˣ + 16 = 0.

Answers

1. Write 64 = 2⁶, so 64^(−5/6) = 2⁻⁵ = 1/32.
2. (3/4)³ = 27/64.
3. √45 = √9 × √5 = 3√5.
4. 9 − 6√2 + 2 = 11 − 6√2.
5. (30 − 10√2)/7.
6. 2^(2x) = 2⁻⁵, so x = −5/2.
7. x = 49.
8. 4 × 1/8 = 1/2.
9. 2√3 + 3√3 = 5√3.
10. 2 + √3.
11. 5^(2x) = 5^(x+3), so 2x = x + 3, giving x = 3.
12. x = 1 or x = 3.

Where marks are usually lost

• Adding indices when the bases differ, or multiplying bases as well as adding indices (2³ × 2⁴ is 2⁷, not 4⁷).
• Reading a negative index as a negative answer instead of a reciprocal.
• Applying the power to the letter but not to the number in a bracket, for example (4x)^(1/2) written as 4x^(1/2).
• Replacing a surd with a rounded decimal in an exact-value question.
• Using the wrong sign in the conjugate, or multiplying only the denominator.
• Expanding (a + √b)² as a² + b, missing the middle term 2a√b.
• Stopping after simplifying and leaving a surd in the denominator when the question says “rationalise”.
• Solving a quadratic in y = aˣ and giving y as the final answer instead of x.
• Skipping lines in a “show that” question. Every step to the given result needs to be visible.

Exam technique

• Write the conjugate you are using before you multiply, so the working is easy to follow.
• Keep surds exact until the final line, and give the form the question asks for.
• Check an index answer by substituting back into one side of the original equation.
• If a question gives a condition such as x > 0, say where you use it.

Official syllabus

AQA, A-level Mathematics 7357, specification (version 1.3, dated 31 January 2018), which applies to A-level exams from June 2018: section 3.3 B: Algebra and functions, content references B1 and B2.