What exponentiation does
Exponentiation is repeated multiplication of a number by itself. When you raise a base to an exponent (or power), you're multiplying that base by itself as many times as the exponent tells you. The exponent can be positive, negative, or even a fraction—each changes the result in a meaningful way.
The formula
result = base ^ exponent (or base to the power of exponent)
For positive integer exponents: base × base × base... (exponent times)
For negative exponents: 1 ÷ (base raised to the positive exponent)
For fractional exponents: the denominator becomes a root, and the numerator is the power
Worked example
Example 1: Positive integer exponent
Raise 2 to the power of 5.
- Base: 2
- Exponent: 5
- Calculation: 2 × 2 × 2 × 2 × 2 = 32
- Result: 32
Example 2: Negative exponent
Raise 3 to the power of −2.
- Base: 3
- Exponent: −2
- Calculation: 1 ÷ (3²) = 1 ÷ 9 = 0.111...
- Result: 0.1111 (or 1/9)
Example 3: Fractional exponent
Raise 16 to the power of 1/2.
- Base: 16
- Exponent: 1/2 (the same as the square root)
- Calculation: √16 = 4
- Result: 4
Example 4: Fractional exponent with numerator > 1
Raise 8 to the power of 2/3.
- Base: 8
- Exponent: 2/3 (cube root, then squared)
- Calculation: (∛8)² = 2² = 4
- Result: 4
Example 5: Decimal exponent
Raise 10 to the power of 0.5.
- Base: 10
- Exponent: 0.5 (equivalent to 1/2)
- Calculation: √10 ≈ 3.162
- Result: 3.162
Understanding different exponent types
| Exponent Type | What Happens | Example |
|---|---|---|
| Positive integer (e.g. 3) | Base multiplies by itself that many times | 2³ = 8 |
| Zero | Any base to the power 0 equals 1 | 5⁰ = 1 |
| Negative (e.g. −2) | Reciprocal of the positive power | 2⁻² = 0.25 |
| Fraction (e.g. 1/2) | Denominator is the root; numerator is the power | 9^(1/2) = 3 |
| Decimal (e.g. 0.5) | Treated as a fraction | 4^0.5 = 2 |
Common mistakes
Confusing base and exponent: 2³ is not the same as 3². Always check which number is the base and which is the power.
Negative base, even exponent: (−2)² = 4 (positive), but −2² = −4 (the negative sign is not squared). The parentheses matter.
Fractional exponents: 4^(3/2) means (√4)³, not √(4³). Work from the root first, then apply the numerator power.
Assuming negative exponents give negative results: They don't. 2⁻³ = 1/8 = 0.125 (positive). A negative exponent flips the fraction, not the sign.
Zero as a base with negative exponent: 0⁻² is undefined because you'd be dividing by zero. Avoid this.
Real-world uses
Exponents appear everywhere: compound interest uses base (1 + rate) raised to the number of periods; scientific notation relies on powers of 10; bacteria growth follows exponential patterns; and half-life calculations in physics and chemistry use negative fractional exponents. Understanding how to compute any exponent—whether it's a whole number, negative, or fractional—is essential for science, finance, and engineering.