Why anything to the zero is 1
We want the rules we already trust to keep working. Take the quotient rule and divide a power by itself: b^n / b^n. By cancellation that is obviously 1. But by the rule it is b^(n−n) = b^0. Both answers must agree, so b^0 = 1 for any nonzero base. The zero exponent isn’t an arbitrary convention — it’s the only value that keeps the algebra consistent.
b^n / b^n = 1 (a thing over itself)
b^n / b^n = b^(n−n) (quotient rule)
= b^0
so b^0 = 1 (b ≠ 0). Examples: 7^0 = 1, (−5)^0 = 1, (2x)^0 = 1Negative exponents are reciprocals
Push the same idea one step further. b^2 / b^5 cancels to 1/b^3, but the rule gives b^(2−5) = b^(−3). So b^(−3) = 1/b^3. In general a negative exponent means take the reciprocal: b^(−n) = 1/b^n. A factor with a negative exponent isn’t “negative” in value — it just belongs on the other side of the fraction bar.
2^(−3) = 1 / 2^3 = 1/8 3x^(−2) = 3 / x^2 (only x moves; the 3 stays) (2/5)^(−1) = 5/2 (flip the fraction) Move a factor across the bar and flip the sign of its exponent: x^(−4) y^2 = y^2 / x^4
Scientific notation
Scientific notation writes a number as a · 10^k, where the decimal coefficient a satisfies 1 ≤ a < 10 and k is an integer. A positive k shifts the point right (big numbers); a negative k shifts it left (small numbers). It’s exactly the negative-exponent idea wearing work clothes.
93,000,000 = 9.3 × 10^7 (move point left 7 places)
0.00042 = 4.2 × 10^(−4) (move point right 4 places)
Multiply: (3 × 10^5)(2 × 10^(−8))
= (3·2) × 10^(5 + (−8))
= 6 × 10^(−3) = 0.006