Every index-law question is really one decision — which of the four laws fits — followed by one quick bit of arithmetic on the indices. Here they are, all on one page.
The Four Laws
Product (multiplying): same base, add the indices. aᵐ × aⁿ = aᵐ⁺ⁿ. Example: 5³ × 5⁴ = 5⁷.
Quotient (dividing): same base, subtract the indices. aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Example: 7⁶ ÷ 7² = 7⁴.
Power of a power: multiply the indices. (aᵐ)ⁿ = aᵐⁿ. Example: (2³)⁴ = 2¹².
Zero index: anything (non-zero) to the power 0 is 1. a⁰ = 1. Example: 9⁰ = 1.
A Worked One
Simplify 2⁵ × 2³ ÷ 2⁴. Same base all the way, so work left to right. Multiplying first: 2⁵ × 2³ = 2⁵⁺³ = 2⁸. Then dividing: 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴. So the whole thing is 2⁴ (which, if you want a plain number, is 16).
Keep the Base, Only Touch the Index
The base never changes — you never multiply 5 × 5 to get 25 in the answer. 5³ × 5⁴ is 5⁷, not 25⁷. The base is the same thing being multiplied; only the count of how many of them (the index) changes.