Laws of Indices (Same Base, Same Indices)

Indices show how many times you multiply a base number by itself. In 2³, the base is 2 and the index is 3. So 2³ means 2 multiplied by itself three times. The laws of indices use this idea to simplify multiplying and dividing calculations.

This lesson covers four laws, organised in two pairs. The first pair works with the same base: you add the indices to multiply and subtract them to divide. The second pair works with the same index: you multiply or divide the bases and keep the index.

Yes, indices can be negative, and the same laws still apply. When you multiply two terms with the same base, you still add the indices. For example, 3⁻⁵ × 3⁷ equals 3 to the power of negative 5 plus 7, which simplifies to 3².

When two numbers share the same index, you combine the bases and keep the index. To multiply, you multiply the bases, so 2³ × 5³ becomes 10³. To divide, you divide the bases, so 6⁵ ÷ 3⁵ becomes 2⁵. The index never changes.

An index counts how many times you multiply the base by itself. So multiplying terms with the same base adds those counts together. In 2³ × 2⁴, three 2s join four 2s. That makes seven 2s, so the indices add to give 2⁷.