Exponent Calculator — Calculate Powers (x^n)
Calculate the power of any number. Supports decimal bases, negative exponents, and shows results in both standard and scientific notation. See also Square Root Calculator and Log Calculator.
Related Calculators:
How to Calculate Exponents
An exponent tells you how many times to multiply a number (the base) by itself. For example, 2^3 means 2 × 2 × 2 = 8. Enter any base and exponent — including decimals and negative numbers — and the calculator will compute the result instantly. Negative exponents produce reciprocals: 2^(-3) = 1/(2^3) = 1/8 = 0.125. Fractional exponents represent roots: 8^(1/3) = ∛8 = 2.
Exponent Formula
x^n = x × x × x × ... (n times)
x^0 = 1 (for any x ≠ 0)
x^(-n) = 1 / x^n
x^(1/n) = ⁿ√x (nth root)
x^(a+b) = x^a × x^b
(x^a)^b = x^(a×b)
Example
2^10 = ?
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= 4 × 4 × 4 × 4 × 4 (grouping pairs)
= 1,024
Frequently Asked Questions
What is any number raised to the power of 0?
Any non-zero number raised to the power of 0 equals 1. This is a mathematical convention that keeps the exponent rules consistent: x^n / x^n = x^(n-n) = x^0 = 1.
What happens with negative exponents?
A negative exponent means "take the reciprocal." So x^(-n) = 1/x^n. For example, 5^(-2) = 1/25 = 0.04.
Can I use decimal exponents?
Yes. A decimal exponent like 2^2.5 is equivalent to 2^(5/2) = 2^2 × 2^(1/2) = 4 × √2 ≈ 5.657. Fractional exponents represent roots combined with powers.
Why does a negative base with a fractional exponent give an error?
In real numbers, a negative base raised to a fractional power (like (-4)^0.5) involves taking an even root of a negative number, which is undefined. The result would be a complex number.
Solved Examples — Exponents
Example: Calculate 3⁴ × 3² using exponent rules
Solution:
Step 1: When multiplying same bases, add exponents: 3⁴ × 3² = 3⁴⁺² = 3⁶
Step 2: Calculate 3⁶ = 729
Answer: 729
Example: Simplify (2³)⁴
Solution:
Step 1: Power of a power rule: multiply exponents: (2³)⁴ = 2³ˣ⁴ = 2¹²
Step 2: Calculate 2¹² = 4,096
Answer: 4,096
Example: A bacteria colony doubles every hour. Starting with 500 bacteria, how many after 8 hours?
Solution:
Step 1: Population = Initial × 2^(hours) = 500 × 2⁸
Step 2: 2⁸ = 256
Step 3: 500 × 256 = 128,000
Answer: 128,000 bacteria
Example: Evaluate 5⁻³
Solution:
Step 1: Negative exponent means reciprocal: 5⁻³ = 1/5³
Step 2: 5³ = 125
Step 3: 1/125 = 0.008
Answer: 1/125 = 0.008
Practice Questions
Try these on your own:
- Calculate 7³ (Answer: 343)
- Simplify 2⁵ × 2³ (Answer: 2⁸ = 256)
- What is 4⁻² ? (Answer: 1/16 = 0.0625)
- Evaluate (3²)³ (Answer: 729)
- If you invest $1,000 at 5% compounded annually, what is it worth after 10 years? Use 1.05¹⁰ (Answer: $1,628.89)
- Simplify 6⁵ ÷ 6² (Answer: 6³ = 216)
Common Mistakes to Avoid
A frequent mistake is confusing the product rule with the power rule. When multiplying same bases, you ADD exponents (2³ × 2⁴ = 2⁷), but when raising a power to a power, you MULTIPLY exponents ((2³)⁴ = 2¹²). Students also incorrectly distribute exponents over addition: (a + b)² ≠ a² + b² — you must expand it as a² + 2ab + b². Another common error involves negative bases: (−3)² = 9, but −3² = −9 (the exponent applies only to 3, not the negative sign, unless parentheses are used). With negative exponents, remember that 2⁻³ = 1/8, not −8. Finally, 0⁰ is a special case that is conventionally defined as 1 in combinatorics but is considered indeterminate in analysis — context matters.
Key Takeaways
- Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ (same base, add exponents).
- Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base, subtract exponents).
- Power rule: (aᵐ)ⁿ = aᵐˣⁿ (raise a power to a power, multiply exponents).
- Zero exponent: a⁰ = 1 for any a ≠ 0.
- Negative exponent: a⁻ⁿ = 1/aⁿ (take the reciprocal).
- Exponents model real-world growth: compound interest, population growth, radioactive decay, and data storage (powers of 2).