Exponent Calculator

How to use the exponent calculator

To find a power (bⁿ): choose “Result of bⁿ”, then enter the base and the exponent. You can type negative numbers, decimals such as 1.5, or fractions such as 2/3 or 1/2.

To find a missing exponent (bˣ = y): choose “Exponent x in bˣ = y”, enter the base and the result, and the calculator solves for x with logarithms.

Whole-number answers up to 15 digits appear in full. Bigger whole numbers are shown in scientific notation with every digit listed underneath, up to 400 digits. Beyond that you get scientific notation and the digit count.

Exponent formula and terms

In bⁿ, b is the base and n is the exponent (also called the power or index). For a positive whole number n:

bⁿ = b × b × … × b (n times)

The definition extends to other exponents:

• Zero: b⁰ = 1 for any b ≠ 0.
• Negative: b⁻ⁿ = 1 ÷ bⁿ.
• Fraction: b^(m/n) = (ⁿ√b)ᵐ, the nth root raised to the power m.
• Decimal: write it as a fraction (0.5 = 1/2) or use b^x = 10^(x × log₁₀ b).

Exponent rules

Rule	Formula	Example
Product rule	aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2⁴ = 2⁷ = 128
Quotient rule	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5⁴ = 5² = 25
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(3²)³ = 3⁶ = 729
Power of a product	(ab)ⁿ = aⁿbⁿ	(2 × 5)³ = 10³ = 1,000
Power of a fraction	(a/b)ⁿ = aⁿ/bⁿ	(1/2)⁴ = 1/16
Negative exponent	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8 = 0.125
Zero exponent	a⁰ = 1	7⁰ = 1
Fractional exponent	a^(m/n) = (ⁿ√a)ᵐ	27^(2/3) = 9

The rules only combine powers with the same base (product and quotient rules) or the same exponent (product and fraction rules). 2³ × 3⁴ cannot be merged into a single power.

Worked examples

Problem	Working	Answer
2¹⁰	ten 2s multiplied	1,024
5⁻²	1 ÷ 25	0.04
8^(2/3)	(∛8)² = 2²	4
2^0.5	√2	1.414214
(−2)³	(−2) × (−2) × (−2)	−8
1.1¹²	twelve 1.1s multiplied	3.138428376721
2ˣ = 1,000	x = 3 ÷ 0.30103	9.965784

The 1.1¹² line is what 10% growth a year does over 12 years: anything growing that fast more than triples. Money compounding at 5% a year doubles when 1.05ˣ = 2, which gives x ≈ 14.206699 years. The compound interest calculator applies the same powers to real deposits, and the CAGR calculator runs the formula backwards to find a growth rate.

Solving for the exponent with logarithms

To solve bˣ = y, take the logarithm of both sides. Because log(bˣ) = x × log(b):

x = log(y) ÷ log(b)

Any base of logarithm works, as long as you use the same one on the top and the bottom. For 3ˣ = 81, x = 1.908485 ÷ 0.477121 = 4, and the calculator confirms 3⁴ = 81 exactly. The base must be positive and not equal to 1, and y must be positive.

Common exponent mistakes

• Brackets around negative bases. −3² means −(3²) = −9, while (−3)² = 9. Type the base as −3 in the calculator to get the second meaning.
• Multiplying instead of raising. 2³ is 2 × 2 × 2 = 8, not 2 × 3 = 6.
• Spreading a power over a sum. (2 + 3)² = 5² = 25, but 2² + 3² = 4 + 9 = 13. Powers spread over products, never over sums.
• Thinking a negative exponent makes a negative number. 2⁻³ = 0.125, which is small but positive.

Very large and very small powers

Powers grow fast. 2¹⁰ is 1,024, 2⁶⁴ has 20 digits, and 2¹⁰⁰⁰ has 302 digits. For whole-number bases and exponents the calculator works with exact integer arithmetic, so every digit is correct, not just the first 15 or 16 that an ordinary calculator keeps. For larger results it switches to scientific notation: 2¹⁰⁰⁰⁰ ≈ 1.995063 × 10³⁰¹⁰. Negative exponents go the other way, making numbers small: 10⁻⁶ is one millionth. To undo a power, use the square root calculator for square, cube and nth roots.