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    Indices, Powers & Standard Form

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    Lesson step 1 of 4Laws of Indices

    Laws of Indices

    The laws of indices let you simplify expressions involving powers. These rules only work when the base is the same.

    Multiplying: aᵐ × an = aᵐ+n
    When you multiply powers with the same base, ADD the indices.
    Example: 𝑥3 × 𝑥5 = 𝑥8

    Dividing: aᵐ ÷ an = aᵐ−n
    When you divide, SUBTRACT the indices.
    Example: y7 ÷ y2 = y5

    Power of a power: (aᵐ)n = aᵐn
    When you raise a power to another power, MULTIPLY the indices.
    Example: (𝑥2)4 = 𝑥8

    Special cases you must memorise:

    a0 = 1 — anything to the power 0 equals 1

    a−n = 1an — a negative power means 'flip it' (put 1 on top and the number underneath)

    a^12 = √a — a power of 12 means square root

    a^13 = cube root of a — a power of 13 means cube root

    a^(m/n) = (nth root of a)ᵐ — root first, then power

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    Key Terms

    Index (power/exponent)
    The small number written above and to the right — it tells you how many times to multiply the base by itself. For example, 2³ means 2 × 2 × 2.
    Standard form
    A neat way to write very big or very small numbers as A × 10ⁿ, where A is between 1 and 10.
    Prime factorisation
    Breaking a number down into the prime numbers that multiply together to make it — like 12 = 2 × 2 × 3.