Year 9: Integer indices

This cheat-sheet covers integer indices. Indices represent how many times a number (the base) is multiplied by itself.

Key concepts

• Positive integer index: xⁿ means x multiplied by itself n times (e.g., 2³ = 2 * 2 * 2 = 8).
• Negative integer index: x^-n is the reciprocal of xⁿ. (e.g., 2^-2 = 1 / 2² = 1 / 4 = 0.25).
• Zero Index:$x^0=1$ (for any non-zero number x).

Rules

• $x^m\times x^n=x^{m+n}$ (Indices with the same base add)
• $x^m/x^n=x^{m-n}$ (Indices with the same base subtract)
• ${\left(x^m\right)}^n=x^{m\times n}$ (Indices multiply)
• $x^0=1$

Examples

Example 1: Simplify $3^2\times 3^4=3^6=729$

Example 2: Simplify $x^5/x^2=x^3$

Example 3: Calculate ${\left(a^2\right)}^3=a^6$

Remember: Always simplify before calculating!