Theory
An index, or power, tells you how many times to multiply a number by itself. In 5³, the base is 5 and the index is 3.
Index laws only work directly when the bases are the same. For example, 3² x 3⁴ can be simplified by adding indices, but 3² x 4² cannot.
When multiplying powers with the same base, add the indices because the repeated multiplications join together.
When dividing powers with the same base, subtract the indices because repeated factors cancel.
When raising a power to another power, multiply the indices. For example, (x³)² means x³ x x³, so it becomes x⁶.
A zero index gives 1 for any non-zero base. A negative index means reciprocal, so a⁻¹ = 1 / a and a⁻² = 1 / a².
Key ruleaᵐ x aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ.