There's a common mistake students make the first time they see 2³. Seeing the small 3 up top, they calculate 2×3=6. But 2³ actually means "multiply 2 by itself 3 times," and the correct answer is 2×2×2=8. Not confusing the number of times you multiply with the size you're multiplying is the first step to understanding exponents.

In the expression aⁿ, the a on the bottom is called the base, and the small n up top is called the exponent. aⁿ means "a multiplied by itself n times." For example, 5⁴ is 5 multiplied by itself 4 times, 5×5×5×5=625, and 10² is 10 multiplied by itself 2 times, 10×10=100.

2 3 = 2 × 2 × 2 = 8 Multiply the base (2) by itself, exponent (3) times ✗ Common mistake 2³ = 2×3 = 6 (wrong) — the exponent is the "number of times"
The exponent shows the "number of times" to multiply, not the size being multiplied

An exponent of 0 is a bit special. Mathematicians defined "multiplying nothing at all" to be 1, by convention (as long as the base isn't 0). Why 1, specifically? A pattern makes it clear. 2⁴=16, 2³=8, 2²=4, 2¹=2 — the value gets cut in half every time the exponent drops by 1. Carry that same pattern forward, and 2⁰ should naturally be half of 2¹=2, which is 1. That's why 2⁰=1 was defined that way.

Once you really understand exponents, the laws of exponents you learn in middle school make a lot more sense. The most basic law is that "multiplying two exponents with the same base adds the exponents" (aᵐ × aⁿ = am+n). Why does that work? Write out 2³ × 2² in full and you get (2×2×2) × (2×2) — and since multiplication doesn't care about order or grouping, you can just chain this into 2×2×2×2×2. How many times did you multiply 2? You combined 3 times with 2 times, for a total of 5 times — in other words, 2⁵. Sure enough, 2³×2²=8×4=32, and 2⁵ is also exactly 32.

This same principle applies to letters just as well as numbers. x³ × x² works the same way — chain (x×x×x)×(x×x) together and you get x⁵. The laws of exponents taught alongside "combining like terms" in middle school math are exactly this idea, extended further. Memorize the laws of exponents as bare formulas and it's easy to get confused once the exponents get complicated, but keep the picture of "chaining together the number of times you multiply" in your head, and you can work out any exponent combination yourself.

On our activity page, you can first change the base and exponent with sliders and see with blocks that an exponent really is a "number of times to multiply," and on the second tab, you can adjust the exponents m and n separately and watch two groups chain together into one to see exactly how aᵐ × aⁿ = am+n holds. The intuition you build here carries straight over into working with algebraic expressions in our follow-up activity on "working with expressions (laws of exponents & like terms)."