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Imagine a computer file doubling in size every hour. If it doubles 3 times this morning and 4 more times tonight, how many total doublings took place?
In math, repeated multiplication uses a base (the value being multiplied) and an exponent (the number of times it multiplies), written as 23β‹…242^3 \cdot 2^4.
In 1637, French mathematician RenΓ© Descartes introduced raised superscript exponents so mathematicians wouldn't have to keep writing out long chains like xβ‹…xβ‹…xx \cdot x \cdot x.
So how do we combine these powers when multiplying them together?

The Product of Powers Rule

When multiplying terms with the same base, keep the base and add the exponents: amβ‹…an=am+na^m \cdot a^n = a^{m+n}.
This rule works because writing out each power simply joins all the repeated factors into one single, longer chain.
This rule relies on the Same-Base Condition: powers can only be combined if their bases are identical. Expressions with different bases, such as x3β‹…y4x^3 \cdot y^4 or 23β‹…342^3 \cdot 3^4, cannot have their exponents added directly.
A crucial requirement is the same-base condition: the rule applies exclusively to expressions with identical bases.
Powers with different bases (such as x3β‹…y4x^3 \cdot y^4 or 23β‹…342^3 \cdot 3^4) cannot have their exponents added directly.
Be careful with hidden exponents: a variable or number written without an explicit exponent has an implicit exponent of 11 (x=x1x = x^1).
This means xβ‹…x4=x1β‹…x4=x1+4=x5x \cdot x^4 = x^1 \cdot x^4 = x^{1+4} = x^5.
Also watch out for hidden exponents: a variable or number written without an exponent has an implicit power of 11 (x=x1x = x^1), meaning xβ‹…x4=x1β‹…x4=x1+4=x5x \cdot x^4 = x^1 \cdot x^4 = x^{1+4} = x^5.