Understanding Exponents and Powers
An exponent (also called a power or index) indicates how many times a number (the base) is multiplied by itself. In the expression xn, x is the base and n is the exponent. Exponents are fundamental in mathematics, science, and many real-world applications.
Positive Exponents
Multiply the base by itself n times
Negative Exponents
Take the reciprocal of the positive power
Fractional Exponents
Represent roots:
Special Cases of Exponents
Zero Exponent Rule
Any non-zero number raised to the power of 0 equals 1.
This comes from the pattern: xn ÷ xn = xn-n = x0 = 1
Exponent of 1
Any number raised to the power of 1 equals itself.
The exponent 1 means “multiply by itself once” which is just the number itself.
Laws of Exponents
| Law | Formula | Example |
|---|---|---|
| Product Rule | xa × xb = xa+b | 23 × 22 = 25 = 32 |
| Quotient Rule | xa ÷ xb = xa-b | 54 ÷ 52 = 52 = 25 |
| Power Rule | (xa)b = xa×b | (32)3 = 36 = 729 |
| Product to Power | (xy)n = xn × yn | (2×3)2 = 22 × 32 = 4 × 9 = 36 |
| Quotient to Power | (x/y)n = xn / yn | (4/2)3 = 43 / 23 = 64/8 = 8 |
| Negative Exponent | x-n = 1/xn | 2-3 = 1/23 = 1/8 |
| Fractional Exponent |
Powers of Common Numbers
Powers of 2
Powers of 10
Perfect Squares
Real-World Applications of Exponents
Compound Interest
A = P(1 + r)n calculates investment growth over time, where money grows exponentially.
Population Growth
P = P0ert models how populations grow exponentially under ideal conditions.
Computer Science
Powers of 2 are fundamental: 210 = 1KB, 220 = 1MB, 230 = 1GB.
Scientific Notation
Large numbers expressed as a × 10n. Speed of light: 3 × 108 m/s.
Physics & Energy
Einstein's E = mc2 shows energy equals mass times the speed of light squared.
Radioactive Decay
N = N0 × (1/2)t/h models how substances decay over time using half-life.
Negative Base with Exponents
When the base is negative, the sign of the result depends on whether the exponent is even or odd:
Even Exponent → Positive
Odd Exponent → Negative
Note: Negative bases with non-integer exponents produce complex numbers, which are beyond real number calculations.
Pro Tip for Students
When solving problems with exponents, always simplify using the laws of exponents before calculating. For example, to find 25 × 23, add the exponents first: 25+3 = 28 = 256. This is much faster than multiplying 32 × 8!