Most tested M4.2

Algebraic Index Laws

Index laws are fundamental rules for simplifying expressions involving powers. Mastering them is essential for manipulating algebraic terms quickly and accurately under timed, no-calculator conditions in the ESAT.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • A power outside a bracket applies to every part of the term inside, including the numerical coefficient.
  • A negative power indicates a reciprocal, e.g., a-n = 1/an. It does not make the term itself negative.
  • A fractional power represents a root, e.g., a^(1/n) is the nth root of 'a'. For a^(m/n), take the nth root first, then apply the power 'm'.
  • Any non-zero base raised to the power of zero equals 1.
    e.g., 50 = 1, x0 = 1
  • Always handle the numerical coefficients separately from the algebraic parts, multiplying or dividing them as required.

Formulae

a^m × an = a^(m+n)

Multiplying terms with the same base; add the indices.

a^m / an = a^(m-n)

Dividing terms with the same base; subtract the indices.

(a^m)n = a^(m*n)

Raising a power to another power; multiply the indices.

a^(-n) = 1 / an

Simplifying terms with a negative index; take the reciprocal to make the index positive.

a^(m/n) = (n-th root of a)^m

Evaluating fractional powers. The denominator 'n' is the root, and the numerator 'm' is the power.

Definitions

Base
The number or variable that is being raised to a power. In the term 5x3, 'x' is the base.
Index / Exponent
The power to which the base is raised, indicating how many times the base is multiplied by itself. In 5x3, '3' is the index.
Coefficient
The numerical factor multiplying a variable term. In 5x3, '5' is the coefficient.

Worked example

Without a calculator, fully simplify the expression (27y-6 / y3)^(2/3).

  1. 1

    First, simplify the terms inside the brackets.

    For the variable 'y', apply the division rule:

    y-6 / y3 = y^(-6 - 3) = y-9
  2. 2

    The expression now is (27y-9)^(2/3).

  3. 3

    Apply the outer power (2/3) to both the coefficient (27) and the variable (y-9) separately.

  4. 4

    For the coefficient:

    27^(2/3).

    First take the cube root (denominator):

    27^(1/3) = 3

    Then square the result (numerator):

    32 = 9
  5. 5

    For the variable:

    (y-9)^(2/3).

    Apply the power-of-a-power rule by multiplying indices:

    y^(-9 × 2/3) = y-6.

  6. 6

    Combine the results to get 9y-6.

    This can also be written as 9 / y6.

Answer: 9y-6 or 9 / y6

Common mistakes

  • ×Sign errors in subtraction: Forgetting that x^a / x^-b becomes x^(a - (-b)) = x^(a+b). This is a frequent source of mistakes.
  • ×Ignoring the coefficient: A common error is to apply a power only to the variable, e.g., incorrectly simplifying (4x2)3 as 4x6 instead of the correct 43 × x6 = 64x6.
  • ×Confusing the rules: Mixing up when to add indices (multiplication) versus when to multiply them (power of a power). Double-check which rule applies.
  • ×Arithmetic slips with roots: Miscalculating roots or powers of coefficients, especially with larger numbers like 64 or 81. Practice is key.

No-calculator tips

  • ✓Know your powers: Instantly recognizing numbers like 8, 27, 64, 125 as perfect cubes, or 16, 81, 625 as perfect fourth powers, saves critical time with fractional indices.
  • ✓Deal with signs first: When an expression involves multiple negative indices, determine the final sign of the index early on to prevent later confusion. e.g., (x-4 / x2)-1 = (x-6)-1 = x6.
  • ✓Break down fractional powers: To evaluate n^(a/b), always do the root part first (b-th root of n), as this makes the number smaller and easier to work with before applying the power 'a'.

Read this topic in the official UAT-UK ESAT guide →

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