Exponent Calculator
Enter a base and an exponent to calculate powers like 2^64, 7^-3, 27^(2/3) or 1.05^10. Whole-number powers of whole numbers are computed exactly, even with thousands of digits; negative exponents are shown as exact fractions; fractional exponents are explained as roots. You can also solve for an unknown exponent, such as 2^x = 1000.
Exponent calculator
Bases can be integers, decimals or fractions (3/4). Exponents can be integers, decimals or fractions (2/3).
Solve for the exponent
Exponents Practice Pack
Laws of exponents reference sheet, powers of 2, 3, 5 and 10 tables, and three exponent worksheets (laws, negative and fractional exponents, scientific notation) with answer keys.
- Laws sheet (PDF)
- Powers table (PDF/XLSX)
- Laws worksheet (PDF, DOCX)
- Negative & fractional (PDF, DOCX)
- Scientific notation (PDF, DOCX)
Formats: PDF, DOCX, XLSX. Instant download after payment (link valid 72 hours, up to 5 downloads). AI-assisted: the templates were drafted with AI help and reviewed and laid out by Kedop.
$3.00 USD, one-time
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What an exponent means
An exponent says how many times to multiply the base by itself: 2^5 = 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the exponent (or power, or index) is 5. Exponents are a compact way to write very large and very small numbers, and they appear everywhere from compound interest and population growth to computer memory sizes and scientific notation.
Laws of exponents
| Rule | Formula | Example |
|---|---|---|
| Product | a^m × a^n = a^(m+n) | 2^3 × 2^4 = 2^7 = 128 |
| Quotient | a^m ÷ a^n = a^(m−n) | 5^6 ÷ 5^4 = 5^2 = 25 |
| Power of a power | (a^m)^n = a^(mn) | (3^2)^3 = 3^6 = 729 |
| Power of a product | (ab)^n = a^n b^n | (2·5)^3 = 8 · 125 = 1000 |
| Zero exponent | a^0 = 1 (a ≠ 0) | 7^0 = 1 |
| Negative exponent | a^(−n) = 1 / a^n | 2^(−3) = 1/8 |
| Fractional exponent | a^(m/n) = (ⁿ√a)^m | 27^(2/3) = 3^2 = 9 |
Negative and fractional exponents
A negative exponent means “one over”: 10^−3 = 1/10^3 = 0.001. A fractional exponent means a root: the denominator is the root and the numerator is the power, so 16^(3/4) = (fourth root of 16)^3 = 2^3 = 8. Negative bases with fractional exponents only give real answers when the root is odd — (−8)^(1/3) = −2, but (−4)^(1/2) is not a real number.
How to use the calculator
- Enter the base: a whole number, decimal or fraction such as 3/4.
- Enter the exponent: a whole number, a negative number, a decimal or a fraction such as 2/3.
- Read the exact result for whole-number exponents, the decimal or scientific form, and the digit count.
- For huge results, the full number appears in the box below — scroll or copy it.
- To find an unknown exponent, use “Solve for the exponent” with a base and a result.
Worked examples
- 2^64 = 18,446,744,073,709,551,616 — 20 digits, the number of values a 64-bit computer word can hold.
- 7^−3 = 1/343 ≈ 0.0029155.
- (3/4)^3 = 27/64 = 0.421875.
- 27^(2/3) = 9, because the cube root of 27 is 3 and 3^2 = 9.
- 1.05^10 ≈ 1.6289 — £1,000 at 5% a year grows to about £1,629 in 10 years.
- 2^x = 1000 gives x = log 1000 ÷ log 2 ≈ 9.9658.
Exact results for big numbers
Ordinary calculators and spreadsheets store numbers with about 15–17 significant digits, so 3^50 appears as 7.17897988e+23 and the last digits are lost. This calculator uses arbitrary-precision integers for whole-number powers, so it can show every digit of 3^50 = 717,897,987,691,852,588,770,249 or even 2^10000, which has 3,011 digits. Decimal and fractional exponents are calculated with standard double-precision arithmetic.
Exponents in everyday life
| Where | Example |
|---|---|
| Computing | 1 KiB = 2^10 = 1,024 bytes; 1 GiB = 2^30 bytes |
| Finance | Compound growth: amount = principal × (1 + rate)^years |
| Science | Speed of light ≈ 3 × 10^8 m/s |
| Biology | Bacteria doubling every 20 minutes: 2^3 = 8 times more in an hour |
| Sound | Every 10 dB is 10 times the intensity |
Scientific notation
Scientific notation writes numbers as a × 10ⁿ with 1 ≤ a < 10. The exponent tells you how many places to move the decimal point: 3.2 × 10⁴ = 32,000 and 3.2 × 10⁻⁴ = 0.00032. Multiplying numbers in scientific notation is easy with the product rule: (2 × 10³)(4 × 10⁵) = 8 × 10⁸. Calculators often show it as 8E+08. The exponent calculator shows results larger than a million or smaller than a millionth in this form.
Exponential growth and decay
| Situation | Formula | Example |
|---|---|---|
| Compound interest | A = P(1 + r)ⁿ | $500 at 4% for 15 years ≈ $900.47 |
| Doubling | N = N₀ × 2^(t/T) | Doubling every 3 years: ×16 after 12 years |
| Half-life decay | N = N₀ × (1/2)^(t/T) | After 3 half-lives, 1/8 remains |
| Rule of 72 | Years to double ≈ 72 ÷ rate % | 6% doubles in about 12 years |
Common mistakes
- −3^2 = −9, but (−3)^2 = 9 — the exponent applies only to what is directly before it.
- a^m × a^n is not a^(m×n); add the exponents instead.
- (a + b)^2 is not a^2 + b^2.
- 0^0 is usually defined as 1 in algebra and computing, but is treated as indeterminate in some contexts.
Privacy
All calculations run in your browser; nothing is uploaded.
Frequently asked questions
How do I calculate an exponent?
Multiply the base by itself as many times as the exponent says, or enter it in the calculator.
What is a negative exponent?
The reciprocal of the positive power: a^(−n) = 1/a^n.
What does an exponent of 1/2 mean?
The square root.
What is any number to the power of 0?
1, for any non-zero base.
How many digits does 2^100 have?
31 digits.
How do I solve 2^x = 1000?
x = log(1000) ÷ log(2) ≈ 9.966.
Why is a^0 equal to 1?
Because a^n ÷ a^n = a^(n−n) = a^0, and any non-zero number divided by itself is 1.
What is 10 to the power of −2?
1/100, or 0.01.
Can the calculator show every digit of 2^1000?
Yes, whole-number powers are exact; 2^1000 has 302 digits.
Is anything uploaded?
No, everything runs locally.