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.
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).