Exponents and powers
A power is a compact way of writing repeated multiplication:
an = a · a · a · ... · a (n factors)
Here, a is the base and n is the exponent.
Important index laws
Multiplication
am · an = am+n
23 · 24 = 27 = 128
Division
am / an = am−n
56 / 52 = 54 = 625
Power of a power
(am)n = am·n
(32)3 = 36 = 729
Power of a product
(a · b)n = an · bn
(2 · 3)3 = 23 · 33 = 216
Zero exponent
a0 = 1 (a ≠ 0)
70 = 1, x0 = 1
Negative exponent
a−n = 1 / an
2−3 = 1/8 = 0.125
Why is a0 = 1? Use the division rule: an / an = an−n = a0. But an / an = 1. Therefore a must0 = 1.
Powers with rational exponents
The exponent can also be a fraction. This gives connections between powers and roots:
Square root
a1/2 = √a
91/2 = √9 = 3
nth root
a1/n = n√a
81/3 = 3√8 = 2
Fractional exponents in general
am/n = n√(am)
82/3 = (3√8)2 = 4
Negative fractional exponent
a−1/2 = 1 / √a
4−1/2 = 1/√4 = 1/2
Roots and powers: A root function such as f(x) = √x is the same as the power function f(x) = x1/2.
Standard form (scientific notation)
Very large or small numbers are often written using powers of ten:
a · 10n (1 ≤ a < 10, n an integer)
- 4 500 000 = 4.5 · 106 (positive n for large numbers)
- 0.000 32 = 3.2 · 10−4 (negative n for small numbers)
- Speed of light: c ≈ 3.0 · 108 m/s
- Mass of an electron: ≈ 9.1 · 10−31 kg
Tip: Count the number of places the decimal point moves. Moving it to the left makes n positive; to the right makes n negative.
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