What does a negative exponent mean?

It flips to the reciprocal. a^(−n) = 1 / a^n. So 2^(−3) is 1/2^3 = 1/8. This isn't arbitrary — it keeps the law a^b × a^c = a^(b+c) consistent across all integer exponents. Specifically, a^1 × a^(−1) = a^0 = 1, which only works if a^(−1) is 1/a. Negative exponents are forced by internal consistency, not invented.

What does a fractional exponent mean?

It's a root in disguise. a^(1/n) means the nth root of a. So a^(1/2) is the square root, a^(1/3) is the cube root, and so on. For mixed forms like a^(p/q), it's the qth root of a^p, or equivalently the qth root of a raised to p. Example: 8^(2/3) means cube root of 8, squared: ∛8 = 2, then 2² = 4. Same answer either way.

Why does my large exponent show 'Infinity'?

JavaScript (which runs this calculator) uses IEEE 754 double-precision floats, which max out at roughly 1.8 × 10^308. Anything beyond that gets represented as 'Infinity' — not actually infinite, just bigger than the format can represent. For genuinely huge calculations (cryptography with 500-digit numbers, for instance), you'd need a big-integer library like Python's or Java's BigInteger.

Is a^b the same as b^a?

Almost never. 2^3 is 8, but 3^2 is 9 — completely different answers. The order of base and exponent matters and they're rarely interchangeable. One beautiful exception: 2^4 = 16 = 4^2, and that's the only positive-integer pair (other than trivial equals like 3^3 = 3^3) where this holds. Mathematical trivia worth keeping in your pocket.

What is exponential growth, and why does it matter?

When a quantity multiplies by a constant factor every time period — compound interest, population growth, viral spread, Moore's Law on chip transistor counts. Small rates compound into shocking long-term changes: 1 percent growth per day doubles in about 70 days; 3 percent per day doubles in just 23. Understanding exponential growth is critical for finance, epidemiology, and any long-term forecasting. Linear intuition fails badly here.

Are exponents related to logarithms?

They're inverse operations. If a^x = y, then log_a(y) = x. That's why logarithmic scales show up everywhere exponential processes are happening — earthquake magnitudes, sound decibels, pH for acidity. The log scale compresses huge ranges into manageable numbers. See the Logarithm Calculator for the flip side of the relationship.

Can I raise a negative number to a fractional exponent?

Only when the root is odd. (−8)^(1/3) = −2 works fine because the cube root of a negative is real and negative. But (−4)^(1/2) has no real value — the square root of a negative is imaginary, and this calculator returns NaN. Pattern: odd roots of negatives are real; even roots of negatives require complex numbers and are out of scope here.

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Exponent / Power Calculator

Math

Raise any base to any exponent — positive, negative, fractional, or decimal — to compute powers, reciprocals, and nth roots for compound interest, scientific notation, and computing.

Math

Exponent / Power Calculator

Generated on May 23, 2026

Base

Exponent

Quick Answer

Raise any base to any exponent — positive, negative, fractional, or decimal — to compute powers, reciprocals, and nth roots for compound interest, scientific notation, and computing.

Overview — Exponent / Power Calculator

Math

Formula

a^b. If b is a positive integer, it's repeated multiplication. a^0 = 1. a^(−b) = 1 / a^b. a^(1/n) is the n-th root of a. a^(p/q) = (a^p)^(1/q).

2^10 = 1,024 = 1 KiB — the foundational unit of binary storage.

2^32 ≈ 4.3 billion — the max value of a 32-bit unsigned integer (and the old IPv4 address limit).

10^100 = 1 googol — the namesake of Google (misspelled on purpose).

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

Compound interest scenario: you're trying to figure out what 5 percent annual growth does to your savings over 20 years. That's 1.05^20, which is about 2.65 — your money roughly two-and-a-half-x's. Or computer science scenario: 2^10 = 1024 (one kibibyte), 2^20 = 1,048,576 (one mebibyte), 2^30 = 1.07 billion (one gibibyte). These powers of 2 are everywhere in computing, from memory sizing to algorithm complexity to IP address space. Exponents are the math that describes 'multiplied by itself n times,' but they extend gracefully to negative exponents (reciprocals), fractional exponents (roots), and even irrational exponents like e or π. The exponent laws — a^b × a^c = a^(b+c), (a^b)^c = a^(b×c), a^(−b) = 1/a^b — keep everything consistent across all these cases. Here's the practical insight people miss: linear intuition fails for exponential growth. If a virus doubles every day and 1 percent of a city is infected today, you might think 'oh, we have a few weeks.' But 0.01 × 2^7 = 1.28, so the whole city is infected in just one week. This is the math behind pandemic projections, viral content spreading, and the Rule of 72 in finance (years to double = 72 / interest rate). Worth respecting.

Pro tips most people learn the hard way

The Rule of 72 is the most useful exponent shortcut for everyday finance. To estimate doubling time at any constant growth rate, divide 72 by the rate. Your money at 6 percent doubles in 12 years (72/6); at 9 percent in 8 years (72/9). It's an approximation of ln(2)/ln(1+r) but accurate enough for mental math.
Powers of 2 deserve memorization for anyone in tech. 2^10 = 1024 (KiB), 2^20 ≈ 1.05 million (MiB), 2^30 ≈ 1.07 billion (GiB), 2^32 ≈ 4.3 billion (the old IPv4 address space limit). Spotting 'this is a power-of-2 boundary' is half of low-level debugging.
Negative exponents are just reciprocals — once that clicks, they stop being scary. 2^(−3) is 1/2^3 = 1/8. Fractional exponents are roots — 8^(1/3) is the cube root of 8, which is 2. Both follow from the same exponent laws as integer exponents; nothing new is happening, just consistent extension.
Exponential growth defeats linear intuition. Pandemic spread, compound interest, viral content, Moore's Law on chip transistors — all exponential. If something doubles every period, after 10 periods you have 1024× the original, after 20 you have over a million times. This is why catching exponential trends early matters far more than reacting to them later.
Honest take: 0^0 is one of math's edge cases. Combinatorics and algebra define it as 1 because that keeps formulas clean (especially the binomial theorem). Analysis sometimes leaves it undefined because the limit depends on the path taken. Pick whichever convention your context demands — but be aware the choice exists.

?What is the Exponent / Power Calculator?

An exponent calculator computes a raised to the power b for any base a and any exponent b — integer, negative, or decimal. Exponents are fundamental to mathematics, science, engineering, and finance. They describe compound growth (money, populations, viral spread), scientific notation (10⁹ for a billion), decay (radioactive half-life), and computer science (binary sizes). Understanding exponents is essential anywhere a quantity is repeatedly multiplied by a constant factor, and this tool handles the computation regardless of whether the exponent is a whole number, a fraction (root), or a negative (reciprocal).

The Formula

a^b. If b is a positive integer, it's repeated multiplication. a^0 = 1. a^(−b) = 1 / a^b. a^(1/n) is the n-th root of a. a^(p/q) = (a^p)^(1/q).

Exponentiation generalizes repeated multiplication. Integer exponents multiply the base by itself that many times; fractional exponents represent roots (a^(1/2) is the square root); negative exponents flip to the reciprocal. All three cases follow from the laws of exponents, especially a^b × a^c = a^(b+c) and (a^b)^c = a^(b×c). These rules keep the algebra consistent across all real exponents, even irrational ones like π.

Powers of Common Bases — Exponent Reference Table

Values of 2^n, 3^n, 10^n, and e^n for common exponents. Essential for computer science, scientific notation, and compound-interest math.

n	2^n	3^n	10^n	e^n (approx)
0	1	1	1	1.00
1	2	3	10	2.72
2	4	9	100	7.39
3	8	27	1,000	20.09
4	16	81	10,000	54.60
5	32	243	100,000	148.41
6	64	729	1,000,000	403.43
8	256	6,561	10^8	2,981
10	1,024 (1 KiB)	59,049	10 billion	22,026
16	65,536	43 million	10^16	8.9 million
20	1,048,576 (1 MiB)	3.5 billion	10^20	485 million
32	4.3 billion	—	10^32	7.9 × 10^13

Put It in Perspective

2^10 = 1,024 = 1 KiB — the foundational unit of binary storage.

2^32 ≈ 4.3 billion — the max value of a 32-bit unsigned integer (and the old IPv4 address limit).

10^100 = 1 googol — the namesake of Google (misspelled on purpose).

Compound growth: (1 + 0.05)^20 ≈ 2.65 — money doubles ~every 14.4 years at 5% interest (Rule of 72 / 5 ≈ 14.4).

Practical Examples

2^10 = 1024 — the classic computing number (1 KiB = 1024 bytes = 2^10).

10^3 = 1000 — the meaning of 'kilo' in the metric system.

2^(−3) = 1/8 = 0.125 — negative exponent flips to a reciprocal.

8^(1/3) = 2 — the cube root of 8, a fractional exponent in action.

(1 + 0.05)^20 ≈ 2.6533 — the factor by which money grows in 20 years at 5% annual compound interest.

e^x is the exponential function that describes continuous growth — central to calculus and probability theory.

Frequently Asked Questions

Any non-zero number raised to the 0 power equals 1 — every time, no exceptions. The reasoning: the exponent law a^(m − n) = a^m / a^n, with m equal to n, gives a^0 = a^m / a^m = 1. The edge case 0^0 is usually defined as 1 in combinatorics and algebra to keep formulas clean, but in analysis it's sometimes left undefined. Pick the convention that fits your context.

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

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Find the LCM and GCD (HCF) of two or more integers using the Euclidean algorithm — useful for simplifying fractions, finding common denominators, and scheduling repeating events.

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Premium Converters

Exponent / Power Calculator

Math

Raise any base to any exponent — positive, negative, fractional, or decimal — to compute powers, reciprocals, and nth roots for compound interest, scientific notation, and computing.

Math

Exponent / Power Calculator

Generated on May 23, 2026

Base

Exponent

Quick Answer

Raise any base to any exponent — positive, negative, fractional, or decimal — to compute powers, reciprocals, and nth roots for compound interest, scientific notation, and computing.

Overview — Exponent / Power Calculator

Math

Formula

a^b. If b is a positive integer, it's repeated multiplication. a^0 = 1. a^(−b) = 1 / a^b. a^(1/n) is the n-th root of a. a^(p/q) = (a^p)^(1/q).

2^10 = 1,024 = 1 KiB — the foundational unit of binary storage.

2^32 ≈ 4.3 billion — the max value of a 32-bit unsigned integer (and the old IPv4 address limit).

10^100 = 1 googol — the namesake of Google (misspelled on purpose).

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

Pro tips most people learn the hard way

The Rule of 72 is the most useful exponent shortcut for everyday finance. To estimate doubling time at any constant growth rate, divide 72 by the rate. Your money at 6 percent doubles in 12 years (72/6); at 9 percent in 8 years (72/9). It's an approximation of ln(2)/ln(1+r) but accurate enough for mental math.
Powers of 2 deserve memorization for anyone in tech. 2^10 = 1024 (KiB), 2^20 ≈ 1.05 million (MiB), 2^30 ≈ 1.07 billion (GiB), 2^32 ≈ 4.3 billion (the old IPv4 address space limit). Spotting 'this is a power-of-2 boundary' is half of low-level debugging.
Negative exponents are just reciprocals — once that clicks, they stop being scary. 2^(−3) is 1/2^3 = 1/8. Fractional exponents are roots — 8^(1/3) is the cube root of 8, which is 2. Both follow from the same exponent laws as integer exponents; nothing new is happening, just consistent extension.
Exponential growth defeats linear intuition. Pandemic spread, compound interest, viral content, Moore's Law on chip transistors — all exponential. If something doubles every period, after 10 periods you have 1024× the original, after 20 you have over a million times. This is why catching exponential trends early matters far more than reacting to them later.
Honest take: 0^0 is one of math's edge cases. Combinatorics and algebra define it as 1 because that keeps formulas clean (especially the binomial theorem). Analysis sometimes leaves it undefined because the limit depends on the path taken. Pick whichever convention your context demands — but be aware the choice exists.

?What is the Exponent / Power Calculator?

The Formula

a^b. If b is a positive integer, it's repeated multiplication. a^0 = 1. a^(−b) = 1 / a^b. a^(1/n) is the n-th root of a. a^(p/q) = (a^p)^(1/q).

Powers of Common Bases — Exponent Reference Table

Values of 2^n, 3^n, 10^n, and e^n for common exponents. Essential for computer science, scientific notation, and compound-interest math.

n	2^n	3^n	10^n	e^n (approx)
0	1	1	1	1.00
1	2	3	10	2.72
2	4	9	100	7.39
3	8	27	1,000	20.09
4	16	81	10,000	54.60
5	32	243	100,000	148.41
6	64	729	1,000,000	403.43
8	256	6,561	10^8	2,981
10	1,024 (1 KiB)	59,049	10 billion	22,026
16	65,536	43 million	10^16	8.9 million
20	1,048,576 (1 MiB)	3.5 billion	10^20	485 million
32	4.3 billion	—	10^32	7.9 × 10^13

Put It in Perspective

2^10 = 1,024 = 1 KiB — the foundational unit of binary storage.

2^32 ≈ 4.3 billion — the max value of a 32-bit unsigned integer (and the old IPv4 address limit).

10^100 = 1 googol — the namesake of Google (misspelled on purpose).

Compound growth: (1 + 0.05)^20 ≈ 2.65 — money doubles ~every 14.4 years at 5% interest (Rule of 72 / 5 ≈ 14.4).

Practical Examples

2^10 = 1024 — the classic computing number (1 KiB = 1024 bytes = 2^10).

10^3 = 1000 — the meaning of 'kilo' in the metric system.

2^(−3) = 1/8 = 0.125 — negative exponent flips to a reciprocal.

8^(1/3) = 2 — the cube root of 8, a fractional exponent in action.

(1 + 0.05)^20 ≈ 2.6533 — the factor by which money grows in 20 years at 5% annual compound interest.

e^x is the exponential function that describes continuous growth — central to calculus and probability theory.

Frequently Asked Questions

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

Browse all Math

GPA Calculator

Calculate your semester or cumulative Grade Point Average on the 4.0 scale, weighted by credit hours, with support for both weighted and unweighted grading systems.

Grade Calculator

Calculate your final grade from weighted categories like tests, homework, and exams.

Student Result Card Calculator

Subject-wise marks, grades, and a printable school result card — ideal for small schools and parents.

Fraction Calculator

Add, subtract, multiply, and divide proper, improper, and mixed fractions with automatic reduction to lowest terms and decimal equivalents for cooking, carpentry, and homework.

Ratio Calculator

Simplify ratios to lowest terms using GCD and solve proportions like a:b = c:x for any missing value — useful for recipes, map scales, aspect ratios, and mixing ingredients.

Average Calculator

Compute the arithmetic mean, median, mode, range, count, sum, minimum, and maximum from any list of numbers — essential for descriptive statistics, surveys, and grading.

LCM / GCD Calculator

Find the LCM and GCD (HCF) of two or more integers using the Euclidean algorithm — useful for simplifying fractions, finding common denominators, and scheduling repeating events.

Prime Number Checker

Test whether any integer is a prime number using trial division up to its square root, and view the complete unique prime factorization for composite numbers.

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See every tool in this category, plus FAQs and category-specific guides.

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Exponent / Power Calculator — Raise any base to any exponent | Premium Converters

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Exponent / Power Calculator

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Overview — Exponent / Power Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Exponent / Power Calculator?

The Formula

Powers of Common Bases — Exponent Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

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Exponent / Power Calculator

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Overview — Exponent / Power Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Exponent / Power Calculator?

The Formula

Powers of Common Bases — Exponent Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

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Exponent / Power Calculator

Exponent / Power Calculator

Overview — Exponent / Power Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Exponent / Power Calculator?

The Formula

Powers of Common Bases — Exponent Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

What is a^0?

What does a negative exponent mean?

What does a fractional exponent mean?

Why does my large exponent show 'Infinity'?

Is a^b the same as b^a?

What is exponential growth, and why does it matter?

Are exponents related to logarithms?

Can I raise a negative number to a fractional exponent?

Popular Conversions

Embed this calculator on your site

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Premium Converters

Exponent / Power Calculator

Exponent / Power Calculator

Overview — Exponent / Power Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Exponent / Power Calculator?

The Formula

Powers of Common Bases — Exponent Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

What is a^0?

What does a negative exponent mean?

What does a fractional exponent mean?

Why does my large exponent show 'Infinity'?

Is a^b the same as b^a?

What is exponential growth, and why does it matter?

Are exponents related to logarithms?

Can I raise a negative number to a fractional exponent?

Popular Conversions

Embed this calculator on your site

More Math Tools

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