Index Laws for Rational Exponents
Laws of indices provide a concise notation and a set of rules for manipulating expressions involving powers, roots, and reciprocals. For the ESAT, they are fundamental for simplifying complex algebraic terms efficiently without a calculator.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- An exponent indicates how many times a 'base' number is multiplied by itself.
- The core rules allow you to combine terms with the same base during multiplication (add powers), division (subtract powers), or when raising a power to another power (multiply powers).
- Negative exponents signify a reciprocal, not a negative number. For example, `a^(-n)` is the same as `1 / an`.
- Fractional exponents correspond to roots. For example, `a^(1/n)` means the n-th root of `a`.
- A common strategy in exam questions is to convert all numerical terms to powers of a common prime base (e.g., rewriting 4, 8, and 16 as powers of 2).
Formulae
a^m × an = a^(m + n) Multiplying terms with the same base.
a^m / an = a^(m - n) Dividing terms with the same base.
(a^m)n = a^(m × n) Raising a power to another power.
a^(-n) = 1 / an Simplifying a term with a negative exponent.
a^(p/q) = (q-throot(a))^p Evaluating a term with a fractional exponent.
a0 = 1 When any non-zero number is raised to the power of zero.
Definitions
- Base
- The number that is being raised to a power. In the term `xn`, `x` is the base.
- Exponent / Index
- The number indicating the power to which the base is raised. In `xn`, `n` is the exponent.
- Rational Exponent
- An exponent that can be written as a fraction `p/q`, where `p` and `q` are integers. This covers integers, fractions, and their negative values.
Worked example
Without using a calculator, simplify the expression (27^(2/3) × 9^(-3/2)) / 3^(-2).
- 1
Identify a common base for all the numbers.
Here, 27 and 9 can both be expressed as powers of 3:
27 = 33 and 9 = 32 - 2
Substitute these into the expression:
((33)^(2/3) × (32)^(-3/2)) / 3^(-2).
- 3
Apply the rule `(a^m)n = a^(m*n)` to the numerator:
3^(3 × 2/3) = 32 and 3^(2 × -3/2) = 3^(-3) - 4
The expression is now (32 × 3^(-3)) / 3^(-2).
- 5
Simplify the numerator using the rule `a^m × an = a^(m+n)`:
3^(2 + (-3)) = 3^(-1).
- 6
The expression simplifies to 3^(-1) / 3^(-2).
- 7
Apply the division rule `a^m / an = a^(m-n)`, being careful with the negative signs:
3^(-1 - (-2)) = 3^(-1 + 2) = 31.
- 8
The final result is 3.
Answer: 3
Common mistakes
- ×Making a sign error during division with negative exponents. Forgetting that `a^m / a^(-n)` becomes `a^(m - (-n)) = a^(m+n)`.
- ×Incorrectly interpreting a negative exponent. `x^(-3)` means `1/x3`, it does not mean `-x3`.
- ×Applying index laws to terms with different bases. For instance, `23 × 32` cannot be simplified by adding the exponents.
No-calculator tips
- ✓When evaluating a fractional power like `64^(2/3)`, always perform the root part first to keep the numbers small. Calculate `cuberoot(64) = 4`, then square the result: `42 = 16`.
- ✓Quickly convert numbers in a question into powers of a common prime base. For example, if you see 4, 8, 32, and 0.5, think of them all as powers of 2: `22`, `23`, `25`, and `2^(-1)`.
- ✓Memorise the first few powers of small integers (e.g., powers of 2 up to 26, powers of 3 up to 34, powers of 5 up to 53) to speed up calculations.