Maths⏱ 5 min read
Powers, Roots, and Exponents: The Complete Guide
Powers and roots appear in compound interest, physics, statistics, and geometry. Here's a clear guide to the rules — including fractional exponents, negative powers, and common mistakes.
Exponents (powers) are shorthand for repeated multiplication. Once you understand the rules, they simplify calculations that would otherwise be unwieldy.
Core Definitions
Base^exponent = base × base × base ... (exponent times)
2^5 = 2 × 2 × 2 × 2 × 2 = 32
3^4 = 3 × 3 × 3 × 3 = 81
10^3 = 1,000
Special cases:
Any number to the power 0 = 1 (e.g. 7^0 = 1)
Any number to the power 1 = itself (7^1 = 7)
0^0 is undefined (mathematical debate ongoing)
The Rules of Exponents
Multiplying same base: add exponents
a^m × a^n = a^(m+n)
2^3 × 2^4 = 2^7 = 128
Dividing same base: subtract exponents
a^m ÷ a^n = a^(m-n)
5^6 ÷ 5^2 = 5^4 = 625
Power of a power: multiply exponents
(a^m)^n = a^(m×n)
(3^2)^4 = 3^8 = 6,561
Power of a product:
(ab)^n = a^n × b^n
(2×3)^3 = 2^3 × 3^3 = 8 × 27 = 216
Negative Exponents
Negative exponent = 1 divided by positive power:
a^(-n) = 1 ÷ a^n
2^(-3) = 1 ÷ 2^3 = 1/8 = 0.125
10^(-2) = 1 ÷ 100 = 0.01
Practical use: scientific notation
3 × 10^(-4) = 3 × 0.0001 = 0.0003
Fractional Exponents and Roots
Fractional exponent = root:
a^(1/n) = nth root of a
a^(1/2) = √a (square root)
a^(1/3) = ∛a (cube root)
8^(1/3) = ∛8 = 2 (because 2^3 = 8)
27^(1/3) = ∛27 = 3
16^(1/4) = 4th root of 16 = 2 (because 2^4 = 16)
Combined: a^(m/n) = (a^m)^(1/n) = (a^(1/n))^m
8^(2/3) = (8^2)^(1/3) = 64^(1/3) = 4
Or: (8^(1/3))^2 = 2^2 = 4 ✓
Square Roots: Key Facts
NumberSquare RootNumberSquare Root
1110010
4214412
9316913
16422515
25525616
36640020
49762525
6481,00031.62
Common Mistakes
WRONG: (a + b)^2 = a^2 + b^2
RIGHT: (a + b)^2 = a^2 + 2ab + b^2
Example: (3 + 4)^2 = 7^2 = 49
NOT: 3^2 + 4^2 = 9 + 16 = 25 ✗
WRONG: √(a^2 + b^2) = a + b
RIGHT: √(a^2 + b^2) is only equal to a+b if a or b = 0