Complete Guide to Exponents
Exponents are a way of representing repeated multiplication. When we write bn, we mean b multiplied by itself n times, where b is the base and n is the exponent (or power).
For example, 23 = 2 × 2 × 2 = 8
Basic Concepts of Exponents
Definition
If b is a real number and n is a positive integer, then:
bn = b × b × b × ... × b (n times)
Special Cases
| Expression | Meaning | Example |
|---|---|---|
| b0 | Any non-zero number raised to the power of 0 equals 1 | 50 = 1 |
| b1 | Any number raised to the power of 1 equals the number itself | 51 = 5 |
| 0n | 0 raised to any positive power equals 0 | 05 = 0 |
| 00 | Undefined or defined as 1 depending on context | - |
| 1n | 1 raised to any power equals 1 | 1100 = 1 |
Examples
24 = 2 × 2 × 2 × 2 = 16
103 = 10 × 10 × 10 = 1,000
(-3)2 = (-3) × (-3) = 9
(-3)3 = (-3) × (-3) × (-3) = -27
Rules of Exponents
1. Product Rule
bm × bn = bm+n
23 × 24 = 23+4 = 27 = 128
x5 × x2 = x7
2. Quotient Rule
bm ÷ bn = bm-n (where b ≠ 0)
27 ÷ 23 = 27-3 = 24 = 16
x8 ÷ x3 = x5
3. Power Rule
(bm)n = bm×n
(23)2 = 23×2 = 26 = 64
(x4)3 = x12
4. Power of a Product
(a × b)n = an × bn
(2 × 3)2 = 22 × 32 = 4 × 9 = 36
(xy)3 = x3y3
5. Power of a Quotient
(a ÷ b)n = an ÷ bn (where b ≠ 0)
(4 ÷ 2)3 = 43 ÷ 23 = 64 ÷ 8 = 8
(x/y)4 = x4/y4
Negative and Fractional Exponents
Negative Exponents
b-n = 1/bn (where b ≠ 0)
2-3 = 1/23 = 1/8 = 0.125
10-2 = 1/102 = 1/100 = 0.01
Fractional Exponents
b1/n = n√b (nth root of b)
bm/n = (n√b)m = n√(bm)
81/3 = 3√8 = 2
161/2 = 2√16 = 4
82/3 = (3√8)2 = 22 = 4
272/3 = (3√27)2 = 32 = 9
Combined Examples
4-1/2 = 1/41/2 = 1/√4 = 1/2 = 0.5
25-3/2 = 1/253/2 = 1/(√25)3 = 1/53 = 1/125 = 0.008
Applications of Exponents
Scientific Notation
Scientific notation expresses a number as a product of a number between 1 and 10 and a power of 10.
5,280 = 5.28 × 103
0.000042 = 4.2 × 10-5
Compound Interest
The formula for compound interest uses exponents:
A = P(1 + r)t
Where: A is the final amount, P is the principal, r is the interest rate, and t is the time period
If you invest $1,000 at 5% annual interest for 3 years:
A = 1000(1 + 0.05)3 = 1000 × 1.053 = 1000 × 1.157625 = $1,157.63
Exponential Growth and Decay
Many natural phenomena follow exponential patterns:
N(t) = N0ekt
Where: N(t) is the quantity at time t, N0 is the initial quantity, k is the growth/decay constant, and t is time
Population growth: If a bacteria population starts with 100 cells and doubles every hour, after t hours:
N(t) = 100 × 2t
After 5 hours: N(5) = 100 × 25 = 100 × 32 = 3,200 bacteria cells
Computing and Data Storage
Powers of 2 are fundamental in computing:
1 Kilobyte (KB) = 210 bytes = 1,024 bytes
1 Megabyte (MB) = 220 bytes = 1,048,576 bytes
1 Gigabyte (GB) = 230 bytes = 1,073,741,824 bytes