Laws of Indices (Higher Tier)

Indices are used to show the power to which a number is raised. For example, in 2^3, the index is 3.
Multiplication Rule: When multiplying two powers with the same base, you add the indices. Example: a^n x a^m = a^(n+m).
Division Rule: When dividing two powers with the same base, you subtract the indices. Example: a^n ÷ a^m = a^(n-m).
Power Rule: When raising a power to a power, you multiply the indices. Example: (a^n)^m = a^(n*m).
Zero Power Rule: Any number raised to the power 0 is 1 (except for 0^0). Example: a^0 = 1.
Negative Power Rule: Any number raised to a negative power is the reciprocal of the number raised to that power. Example: a^-n = 1/a^n.
The Root Rule: The nth root of a number b can be expressed as b^(1/n). Example: √a = a^(1/2).

Understand how to apply the laws of indices to simplify expressions. This includes knowing when to add, subtract or multiply indices.
Get used to working with powers and roots in the same equation, as they often appear in the same question.
Creating equalities using the laws of indices. This might include solving for unknowns in the powers or equalising two expressions.

Be comfortable with the basics of index notation, including positive, negative, fractional indices, and zero index.
Applying the laws of indices to simplify an algebraic expression is a key skill for any mathematics exam.
Although it’s tempting to apply the laws of indices to bases that aren’t the same, this is a common pitfall to avoid.
Practice simplifying and manipulating indices to get the hang of using these laws - indices often come up in a variety of mathematical problems, so they’re an important tool to have at your disposal.