Introduction to Exponents
Repeated multiplication, and the two exponents that look strange at first.
The explanation
An exponent is a counter for repeated multiplication. 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The small number counts the factors, it does not multiply.
That is the biggest early trap: 2⁵ is not 10.
Two special cases. Anything to the first power is itself: 7¹ = 7. Anything (except 0) to the zero power is 1: 7⁰ = 1. That second one looks arbitrary until you look at a pattern: 2³ = 8, 2² = 4, 2¹ = 2. Each step down halves the result, so the next one has to be 1.
For a positive integer n, aⁿ is defined as the product of n copies of a, with a¹ = a as the base case and the recursive rule aⁿ = a·aⁿ⁻¹.
The value of a⁰ is then forced rather than assumed. The quotient rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ applied to aⁿ ÷ aⁿ gives a⁰ on one side and 1 on the other, so a⁰ = 1 for every a ≠ 0. The case 0⁰ is left undefined in this context because the two natural extensions disagree.
Watch the interaction with the unary minus. Exponentiation binds tighter than negation, so −a² means −(a²) while (−a)² means the square of the negative. For a = 5 those are −25 and 25 respectively, and no amount of care about signs elsewhere will save work that gets this wrong.
Worked example
Evaluate 3⁴, (−2)³, and −2⁴.
- 3⁴ = 3·3·3·3 = 81.
- (−2)³ = (−2)(−2)(−2) = −8 (odd number of negatives).
- −2⁴ = −(2·2·2·2) = −16 (the minus is not inside the power).
Answer: 81, −8, −16
Common mistakes
- Multiplying base by exponent: reading 4³ as 12 instead of 64.
- Assuming (−3)² and −3² are the same. They are 9 and −9.