AQA · GCSE · Mathematics

    Course version: 8300 · Version from Sept 2015

    Structure and calculation: calculate with fractional indices (Higher tier only)

    This Higher tier row extends index work to fractional indices.

    • 3 key terms

    Study notes

    From the checked teaching notes for this specification point.

    The idea

    This Higher tier row extends index work to fractional indices. A unit fraction index means a root: a^(1/n) is the nth root of a, so 25^(1/2) = 5 and 27^(1/3) = 3. A general fractional index combines a root and a power: a^(m/n) means the nth root of a raised to the power m, equivalently the mth power of the nth root. So 8^(2/3) = (∛8)² = 2² = 4, and 16^(3/4) = (⁴√16)³ = 2³ = 8. Negative fractional indices combine a reciprocal with a root and power: 27^(−1/3) = 1/∛27 = 1/3. Take the root first when the numbers are friendly, since the intermediate value stays small. In a multiple-choice question, evaluate fully before comparing options.

    What earns the marks

    • Interprets a unit fraction index as a root, for example 64^(1/2) = 8 and 64^(1/3) = 4.
    • Interprets a general fractional index as root then power, for example 27^(2/3) = (∛27)² = 3² = 9.
    • Evaluates fractional indices where the numerator is not 1, for example 16^(3/4) = (⁴√16)³ = 2³ = 8.
    • Handles negative fractional indices as reciprocals, for example 8^(−2/3) = 1/(∛8)² = 1/4.
    • Chooses the root-first order to keep arithmetic exact, for example evaluating 32^(2/5) as (⁵√32)² = 2² = 4.
    • Selects the single correct option after independent evaluation rather than by pattern-matching the options.

    Common mistakes and how to put them right

    • Multiplying the base by the fraction.

      Put it right: A unit fraction index is a root, so 9^(1/2) = 3.

    • Swapping the root and the power.

      Put it right: The denominator gives the root and the numerator gives the power, so 8^(2/3) = (∛8)² = 2² = 4.

    • Ignoring the negative sign in a fractional index.

      Put it right: The negative sign means a reciprocal, so 4^(−1/2) = 1/√4 = 1/2.

    • Taking the power first with large numbers and losing accuracy.

      Put it right: Take the root first, so (∛27)² = 3² = 9.

    Exam tips

    • Rewrite a^(m/n) as (ⁿ√a)ᵐ before calculating, and take the root first when the radicand is a perfect power.
    • For negative fractional indices, write the reciprocal first, for example 16^(−3/4) = 1/(⁴√16)³, then evaluate.
    • Check your answer by raising it to the reciprocal power, for example confirming 4^(3/2) = 8 because 8^(2/3) = 4.
    • Keep exact values rather than decimals so the option you choose matches the exact surd or integer given.

    Try the lesson

    The start of the Structure and calculation: calculate with fractional indices (Higher tier only) lesson, made from checked teaching notes for this specification point.

    Have a go first

    Work out the value of 49^(1/2).

    Listen

    This Higher tier row extends index work to fractional indices. A unit fraction index means a root: a^(1/n) is the nth root of a, so 25^(1/2) = 5 and 27^(1/3) = 3. A general fractional index combines a root and a power: a^(m/n) means the nth root of a raised to the power m, equivalently the mth power of the nth root. So 8^(2/3) = (∛8)² = 2² = 4, and 16^(3/4) = (⁴√16)³ = 2³ = 8. Negative fractional indices combine a reciprocal with a root and power: 27^(−1/3) = 1/∛27 = 1/3. Take the root first when the numbers are friendly, since the intermediate value stays small. In a multiple-choice question, evaluate fully before comparing options. A strong answer earns marks for these points. Interprets a unit fraction index as a root, for example 64^(1/2) = 8 and 64^(1/3) = 4. Interprets a general fractional index as root then power, for example 27^(2/3) = (∛27)² = 3² = 9. Evaluates fractional indices where the numerator is not 1, for example 16^(3/4) = (⁴√16)³ = 2³ = 8. Handles negative fractional indices as reciprocals, for example 8^(−2/3) = 1/(∛8)² = 1/4. A common slip: Multiplying the base by the fraction: the learner computes 9^(1/2) as 9 × 1/2 = 4.5. Instead, a unit fraction index is a root, so 9^(1/2) = 3. Exam tip: Rewrite a^(m/n) as (ⁿ√a)ᵐ before calculating, and take the root first when the radicand is a perfect power.

    See it

    A flowchart starting from a node labelled "Fractional index a^(m/n)". Two arrows lead from it: one to "Denominator n gives the root" and one to "Numerator m gives the power". Both of these lead to "Take the root first: ⁿ√a", which leads to "Raise the result to the power m", which leads to "Evaluate fully before comparing options". A separate arrow from the starting node leads to "Negative sign means a reciprocal", which leads to "Write 1 over the positive fractional power", which then leads into "Take the root first: ⁿ√a".
    Shows how a fractional index splits into a root given by the denominator and a power given by the numerator, with a negative sign meaning a reciprocal.

    Key words

    Say what each word means before you turn the card.

    unit fraction index

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    general fractional index

    Try answering aloud or in your head before turning the card.

    root-first evaluation

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    See every definition
    unit fraction index
    A unit fraction index means a root: a^(1/n) is the nth root of a.
    general fractional index
    A general fractional index combines a root and a power: a^(m/n) means the nth root of a raised to the power m, equivalently the mth power of the nth root.
    root-first evaluation
    Take the root first when the numbers are friendly, since the intermediate value stays small.

    Label the diagram

    Some boxes have lost their names. Pick the right name for each number.

    The diagram again, with 3 boxes numbered instead of named. A flowchart starting from a node labelled "Fractional index a^(m/n)". Two arrows lead from it: one to "Denominator n gives the root" and one to "Numerator m gives the power". Both of these lead to "Take the root first: ⁿ√a", which leads to "Raise the result to the power m", which leads to "Evaluate fully before comparing options". A separate arrow from the starting node leads to "Negative sign means a reciprocal", which leads to "Write 1 over the positive fractional power", which then leads into "Take the root first: ⁿ√a".

    • 1Box 1

    • 2Box 2

    • 3Box 3

    0 of 3 labelled

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