Learning objectives
Apply the laws of indices to positive, negative, zero and fractional exponents, moving freely between root form and index form
Simplify surds by extracting square factors, and add, subtract and multiply surd expressions exactly
Rationalise a denominator containing a single surd or a binomial surd, using the conjugate, and give the answer in the form a + b√c
Solve equations in which the unknown appears as an exponent by writing both sides as powers of one base and equating the indices
Key definitions
Index (exponent)
The power to which a base is raised: in aⁿ the number a is the base and n is the index. The index laws hold for every rational value of n.
Surd
A root such as √2 or ∛5 whose exact value is irrational, so it is left in root form rather than written as a rounded decimal.
Simplified surd
A surd written as k√m where m has no square factor other than 1, for example √48 = 4√3.
Rationalising the denominator
Rewriting a fraction so that no surd remains in the denominator, by multiplying the numerator and the denominator by the same quantity.
Conjugate
The partner of a + b√c is a − b√c. Their product a² − b²c is rational, which is what clears a binomial surd from a denominator.
Exact form
An answer left as an integer, a fraction, a surd or a multiple of π, with no rounding. Paper 1 asks for this.
Explanation
The laws of indices. Every manipulation in this topic comes from three rules: aᵐ × aⁿ = a^(m + n), aᵐ ÷ aⁿ = a^(m − n) and (aᵐ)ⁿ = a^(mn). They hold for any rational indices provided the base a is the same in each term, so the first move in almost every question is to rewrite the bases so that they match. Two special cases follow at once: putting m = n in the division law gives a⁰ = 1 for any non-zero a, and putting m = 0 gives a⁻ⁿ = 1/aⁿ. A negative index therefore means "take the reciprocal", never "make the answer negative": 4⁻² = 1/16.
Fractional indices and root form. The rule (aᵐ)ⁿ = a^(mn) forces a^(1/n) to be the nth root of a, because (a^(1/n))ⁿ = a¹ = a. Putting the two parts together, a^(m/n) = (ⁿ√a)ᵐ: the denominator is the root and the numerator is the power. Take the root first, so the numbers stay small: 32^(3/5) = (⁵√32)³ = 2³ = 8, and 8^(−2/3) = 1/(∛8)² = 1/4. Travelling the other way, √x = x^(1/2), ∛(x⁵) = x^(5/3) and 1/x⁴ = x⁻⁴, and it is this index form that later lets you differentiate or integrate such expressions.
Simplifying and combining surds. To simplify a surd, take out the largest square factor: √48 = √16 × √3 = 4√3 and √200 = √100 × √2 = 10√2. Only like surds can be added or subtracted, so simplify before collecting: √50 − √18 + √8 = 5√2 − 3√2 + 2√2 = 4√2. Multiplication uses √a × √b = √(ab), and brackets expand exactly as in ordinary algebra: (3 + 2√5)(1 − √5) = 3 − 3√5 + 2√5 − 10 = −7 − √5. Note that √a + √b is never the same as √(a + b); the single counterexample √9 + √16 = 7 against √25 = 5 settles that for good.
Rationalising the denominator. An exact answer is expected to carry no surd in the denominator. With a single surd, multiply the top and the bottom by that surd: 6/√3 = 6√3/3 = 2√3. With a binomial surd such as 2 − √3, multiply the top and the bottom by the conjugate 2 + √3, because (a + √b)(a − √b) = a² − b is rational. So 1/(2 − √3) = (2 + √3)/(4 − 3) = 2 + √3. Multiply the numerator by the same factor every time, expand with care, and present the result in the form the question asks for, usually a + b√c.
Equations with the unknown in the index. If both sides can be written as powers of one base then the indices can be equated, since aˣ = aʸ forces x = y whenever a is positive and a ≠ 1. For 4^(x + 1) = 8^(x − 1), write 4 = 2² and 8 = 2³ to obtain 2^(2x + 2) = 2^(3x − 3), so 2x + 2 = 3x − 3 and x = 5. An equation such as 2^(2x) − 9(2ˣ) + 8 = 0 is a hidden quadratic: put u = 2ˣ, so u² − 9u + 8 = 0 gives u = 1 or u = 8, and therefore x = 0 or x = 3. When the bases genuinely cannot be matched, logarithms are the tool instead.
Exact answers on the non-calculator paper. Paper 1 is taken without a calculator, so a surd, a fraction or a multiple of π is the finished answer and a rounded decimal is not. Keep √2, √3 and √5 intact all the way through the working rather than converting near the end, then check that the denominator is rational and that every surd is fully simplified. The habit pays off repeatedly: exact surds reappear in the trigonometric ratios for π/6, π/4 and π/3, in coordinate geometry lengths such as √20 = 2√5, and inside the discriminant of a quadratic.
Formulae
aᵐ × aⁿ = a^(m + n); aᵐ ÷ aⁿ = a^(m − n); (aᵐ)ⁿ = a^(mn)
a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ; (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ/bⁿ
a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
√a × √b = √(ab); √a ÷ √b = √(a/b), for a ≥ 0 and b greater than 0
(a + √b)(a − √b) = a² − b
If aˣ = aʸ with a positive and a ≠ 1, then x = y
Worked examples
q: Express (5 + √3)/(2 − √3) in the form a + b√3, where a and b are integers.
steps: The denominator is a binomial surd, so multiply the numerator and the denominator by the conjugate 2 + √3.
Numerator: (5 + √3)(2 + √3) = 10 + 5√3 + 2√3 + 3 = 13 + 7√3.
Denominator: (2 − √3)(2 + √3) = 2² − (√3)² = 4 − 3 = 1.
The expression is (13 + 7√3)/1 = 13 + 7√3, so a = 13 and b = 7.
Check the size: √3 ≈ 1.73, so the answer is about 25, and (5 + 1.73)/(2 − 1.73) ≈ 6.73/0.27 ≈ 25 as well.
In context
Compound interest, population growth and radioactive decay are all written as a base raised to a power, so the index laws are what convert a yearly rate into a monthly one or a half-life into a decay factor.
Surds keep construction and design work exact: the diagonal of a one-metre square is √2 metres, and rounding that figure at every step of a tiled run accumulates a visible error.
Common misconceptions
Students often write √(a + b) as √a + √b. In fact a root cannot be split across a sum: √(9 + 16) = √25 = 5, not 3 + 4 = 7.
Students often read a negative index as a negative answer. In fact 4⁻² = 1/4² = 1/16, which is positive; the minus sign only instructs you to take the reciprocal.
Students often multiply the denominator by the conjugate and forget the numerator. In fact both must be multiplied by the same quantity, or the value of the fraction is changed.
Students often treat 8^(2/3) as 8 × 2 ÷ 3. In fact the denominator gives the root and the numerator gives the power, so 8^(2/3) = (∛8)² = 2² = 4.
Study tips
Take the root before the power when evaluating a fractional index by hand: 27^(4/3) = (∛27)⁴ = 3⁴ = 81, which is far easier than finding 27⁴ first.
Simplify every surd to the form k√m before adding or subtracting, because only like surds collect together.
On Paper 1 leave √2, √3 and π in the final line; a decimal is not an exact answer and loses the accuracy mark.
For an index equation, say out loud which common base you are choosing before you rewrite; picking 2 for a question mixing 4, 8 and 16 saves a whole line of work.
Reasoning
Negative and fractional indices are the form an expression must be put into before the power rule for differentiation can be applied later in this course.
Summary
Indices and surds are the exact-arithmetic toolkit that the non-calculator paper is built on. The three index laws aᵐ × aⁿ = a^(m + n), aᵐ ÷ aⁿ = a^(m − n) and (aᵐ)ⁿ = a^(mn) extend to zero, negative and fractional powers, where a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(m/n) = (ⁿ√a)ᵐ, so root form and index form are two ways of writing the same thing and you should take the root before the power when working by hand. A surd is simplified by extracting its largest square factor, like surds are then collected, and products expand as ordinary brackets using √a × √b = √(ab). A denominator is rationalised by multiplying the top and bottom by the surd itself, or by the conjugate when the denominator is a binomial, since (a + √b)(a − √b) = a² − b is rational. When the unknown sits in the index, rewrite both sides with a common base and equate the indices, spotting the hidden quadratic when a term such as 2^(2x) appears alongside 2ˣ. Above all, leave answers exact: surds, fractions and multiples of π are the finished form on Paper 1.
Check your understanding
Question 1: Simplified, √75 is…
1. 3√5
2. 25√3
3. 5√3
Question 2: The value of 16^(−3/4) is…
1. 8
2. 1/8
3. −8
Question 3: Which expression rationalises the denominator of 1/(4 − √7)?
1. 4 + √7
2. 4 − √7
3. √7