How Multiplying Exponents Works
Multiplying two exponential expressions means multiplying two powers, such as 6⁵ × 6². When the powers share the same base, there is a shortcut called the product rule: instead of computing each power in full and then multiplying, you can simply add the exponents. When the bases are different, there is no shortcut — you must evaluate each power on its own and then multiply the two numbers directly. This calculator does both automatically and shows each term along with the final product.
The product rule for exponents
For a nonzero base a and integers m and n, a^m × a^n = a^(m+n). This works because a^m is a multiplied by itself m times, and a^n is a multiplied by itself n times; multiplying them together simply strings all those factors of a in a row, giving a multiplied by itself (m + n) times. For example, 6⁵ × 6² = (6·6·6·6·6) × (6·6) = 6·6·6·6·6·6·6 = 6⁷ = 279,936, which matches 6^(5+2) = 6⁷. If adding the exponents gives 0, the rule a⁰ = 1 applies (for a ≠ 0). If it gives a negative number, the result is a fraction: a^(−n) = 1 / a^n.
When the bases don't match
The product rule only simplifies powers that share the same base. If you're multiplying, say, 6⁵ × 2², there is no exponent shortcut — you evaluate each power separately (6⁵ = 7,776 and 2² = 4) and then multiply the results (7,776 × 4 = 31,104). This calculator always computes each term this way, and additionally shows the simplified a^(m+n) form whenever the two bases match.
Common mistakes
- Multiplying the exponents instead of adding them: a^m × a^n is a^(m+n), not a^(m×n) — that is a different operation (a power of a power, (a^m)^n).
- Applying the rule with different bases: 6⁵ × 2² cannot be simplified to 12^something; you must evaluate each power first.
- Multiplying the bases when the exponents match instead: that rule is different — a^n × b^n = (ab)^n only applies when the exponents are equal, not the bases.
- Assuming a negative base with a fractional exponent is real: an expression like (−9)^0.5 involves the square root of a negative number and is not a real number.
Applications
- Simplifying algebraic expressions and polynomial products in algebra and precalculus.
- Working with scientific notation, where multiplying measurements often means multiplying powers of 10.
- Computer science and information theory, where multiplying powers of 2 describes memory, addressing, and data-rate relationships.
- Compound growth comparisons, where products of the same base raised to different periods reduce to a single power.