02 October 2025
Indices are an essential foundation that students need as part of their preparation for the O Level Maths paper. The Laws of Indices spell out the rules behind determining how indices work when applied in various mathematical scenarios, and this handy article will outline everything you need to master them.
Index numbers (or indices, which is the plural form of index) refer to the powers, or exponents, of numbers. A power indicates how many times a number is multiplied by itself, and is expressed as such:
xa
In the above example, “x” is the base, while the smaller “a” is the power or exponent. The base is the number being raised to the power, multiplying itself by “a” times. As such, 22 is essentially 2 × 2, and the answer is 4.
The Laws of Indices outline how students should approach mathematical expressions that involve indices. There are six Laws of Indices covering varying scenarios.
Any number that has an power of one is itself. As such:
21 = 2
While it is not one of the Laws of Indices, this elementary rule is still fundamental to learning about indices.
If a number is raised to the index of zero, then the answer is simply 1.
20 = 1
When multiplying two numbers with similar bases but different powers, you can proceed to add the powers and raise them to that same base.
xa × xb = xa + b
To illustrate this with a sample:
23 × 25 = 23 + 5 = 28
Conversely, when dividing two numbers of similar bases and different powers, you will subtract the second power from the first power.
xa ÷ xb = xa – b
Here is a simple example:
26 ÷ 23 = 26 – 3 = 23
When multiplying two different bases that have the same indices, you will need to multiply both bases and raise the index to the new sum.
xa × ya = (x × y)a
For example:
23 × 33 = (2 × 3)3 = 63
Conversely, when dividing two different bases with the same indices, you will divide both bases and raise the index to this new sum. You can express this division of bases in fractions.
xa ÷ ya = (x⁄y)a
For example:
33 ÷ 53 = (3⁄5)3
If there is a power outside a bracket containing a base with a different power, you can multiply both powers together like so:
(xa)b = xab
For example:
(22)3 = 22 × 3 = 26
If, however, there are two numbers in the bracket without a power, it is instead solved like this:
(xy)a = xaya
To make this clearer, consider the following example:
(3x)2 = 32x2 = 9x2
A negative power is the reciprocal: the numerator is one and the denominator is the base with a positive power, like so:
x–a = 1 / xa
If a fraction has an power, you will apply the power to both the numerator and the denominator.
(x / y)a = xa / ya
If a fraction has a negative power, then it becomes the reciprocal of the fraction to the positive power, as illustrated in the following example:
(x / y)–a = (y / x)a = ya / xa
Should you encounter a base number with a fraction index, then the denominator becomes the root of the number or letter:
x1/a = a √ x
Meanwhile, a negative, fractional power is one over a root:
x–1/a = (x1/a)–1 = 1 / a √ x
Understanding the laws of indices is a key part of mathematical learning. Not only does it teach students how to simplify the multiplication of a number by itself over a certain number of times, but it also serves as an entry point to more complex algebraic expressions. More specifically, they are instrumental in ensuring students understand and solve questions on differentiation and integration. Mastering this fundamental set of mathematical principles will help students do well in the more advanced O Level Maths topics that follow.
At Matrix Math, we can guide students to success with personalised tutoring to help them apply the laws of indices correctly. We use valuable revision materials to help them further their understanding and strengthen their mathematical foundation. Contact us today to discover how we can prep students towards success.
