Laws of Indices
Index laws are the fundamental rules for manipulating numbers written with powers (indices). Mastering them is crucial for simplifying complex numerical expressions quickly and accurately without a calculator, a core skill for 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
- When terms with the same base are multiplied or divided, their indices are added or subtracted, respectively.
- A negative power indicates a reciprocal; for example, x-n is equivalent to 1/(xn).
- A fractional power represents a root; for instance, x^(1/n) is the n-th root of x.
- When a power is raised to another power, such as (x^a)^b, the indices are multiplied together to give x^(ab).
- Any non-zero number raised to the power of zero is equal to 1.
- When simplifying expressions, aim to convert all numbers to powers of a common base where possible (e.g., express 4, 8, and 32 as powers of 2).
Formulae
a^m × an = a^(m+n) To multiply two terms that have the same base.
a^m / an = a^(m-n) To divide two terms that have the same base.
(a^m)n = a^(m*n) To raise an existing power to a further power.
a^(-n) = 1 / an To handle negative indices by converting them into their positive reciprocal form.
a^(m/n) = (n-th root of a)^m To evaluate fractional indices, which represent a combination of a root and a power.
a0 = 1 A fundamental identity for any non-zero base 'a' raised to the power of zero.
Definitions
- Base
- The number that is being repeatedly multiplied in an expression involving a power. In 53, the base is 5.
- Index / Exponent
- The number indicating how many times the base is to be multiplied by itself. In 53, the index is 3.
- Reciprocal
- The result of dividing 1 by a number. For example, the reciprocal of x is 1/x. This is key to understanding negative indices.
Worked example
Without using a calculator, evaluate the expression: (125^(2/3) × 5-1) / (251.5)
- 1
First, express all numbers as powers of a common base, which is 5 in this case.
125 = 53 and 25 = 52 - 2
Substitute these into the expression:
((53)^(2/3) × 5-1) / ((52)1.5).
- 3
Simplify the powers using the law (a^m)n = a^(m*n).
(53)^(2/3) becomes 5^(3 × 2/3) = 52.
And (52)1.5 becomes 5^(2 × 1.5) = 53.
- 4
The expression is now:
(52 × 5-1) / 53.
- 5
Simplify the numerator using the law a^m × an = a^(m+n):
52 × 5-1 = 5^(2 + (-1)) = 51 - 6
The expression simplifies to 51 / 53.
- 7
Finally, use the division law a^m / an = a^(m-n):
5^(1-3) = 5-2 - 8
Convert the negative power to a fraction:
5-2 = 1 / 52 = 1/25
Answer: 1/25
Common mistakes
- ×Confusing the multiplication rule with the power-of-a-power rule: adding indices (for a^m × an) versus multiplying them (for (a^m)n).
- ×Incorrectly handling negative indices. Forgetting that a-n means 1/an and not -an.
- ×Applying the root and power in the wrong order for fractional indices. For 27^(2/3), it's much easier to find the cube root first (3) then square it (9), than to square 27 first.
- ×Assuming (a + b)n is equal to an + bn. Index laws only apply to multiplication and division, not addition or subtraction.
No-calculator tips
- ✓Memorise the squares of numbers up to 15, cubes up to 5, and the first few powers of 2 and 3. This makes recognising roots and bases much faster.
- ✓When faced with a fractional power like x^(m/n), always calculate the n-th root of x first. This makes the numbers smaller and easier to manage before you apply the power of m.
- ✓If an expression contains multiple bases (e.g., 4, 8, 16), immediately look for a common base to rewrite them in (e.g., 22, 23, 24). This is often the key to simplification.