Exponent Calculator
X raised to the power of Y
Enter base and exponent below
Supports negative and decimal exponents
Quick examples
Enter a base and exponent to calculate X raised to the power of Y.
Result
Details
Base (X)
Exponent (Y)
Result
Scientific notation
Reciprocal (1/result)
Exponent type
Exponents Explained
An exponent tells you how many times to multiply the base by itself: 210 means ten 2s multiplied together, which is 1,024. A negative exponent gives a reciprocal (2−3 = 1/8 = 0.125) and a fractional one gives a root (90.5 = 3). Enter any base and exponent above, including decimals and negatives, to get the result, scientific notation and reciprocal.
An exponent indicates how many times a base number is multiplied by itself. In 2^10, the base is 2 and the exponent is 10, meaning 2×2×2×2×2×2×2×2×2×2 = 1,024. Exponents follow specific laws that make complex calculations manageable.
Negative Exponents
X^(−n) = 1 / X^n. Example: 2^(−3) = 1/8 = 0.125. Negative exponents represent reciprocals. 10^(−2) = 1/100 = 0.01.
Zero Exponent
Any non-zero base raised to the power of 0 equals 1. X^0 = 1 for all X ≠ 0. Example: 5^0 = 1, 1000^0 = 1, (−7)^0 = 1.
Fractional Exponents
X^(1/n) = nth root of X. X^(1/2) = √X. Example: 9^(1/2) = 3. 8^(1/3) = 2. 27^(1/3) = 3. X^(m/n) = (nth root of X)^m.
Large Exponents
2^10 = 1,024. 2^20 ≈ 1 million. 2^30 ≈ 1 billion. 2^32 = 4,294,967,296 (the number of values a 32-bit integer can hold). Results shown in scientific notation above 10^15.
Powers Quick Reference Table
Exact values for the bases people look up most often.
| n | 2n | 3n | 5n | 10n |
|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 |
| 1 | 2 | 3 | 5 | 10 |
| 2 | 4 | 9 | 25 | 100 |
| 3 | 8 | 27 | 125 | 1,000 |
| 4 | 16 | 81 | 625 | 10,000 |
| 5 | 32 | 243 | 3,125 | 100,000 |
| 6 | 64 | 729 | 15,625 | 1,000,000 |
| 7 | 128 | 2,187 | 78,125 | 10,000,000 |
| 8 | 256 | 6,561 | 390,625 | 100,000,000 |
| 9 | 512 | 19,683 | 1,953,125 | 1,000,000,000 |
| 10 | 1,024 | 59,049 | 9,765,625 | 10,000,000,000 |
| 12 | 4,096 | 531,441 | 244,140,625 | 1,000,000,000,000 |
Negative exponents
| n | 2n | 3n | 10n |
|---|---|---|---|
| −1 | 1/2 = 0.5 | 1/3 = 0.3333 | 0.1 |
| −2 | 1/4 = 0.25 | 1/9 = 0.1111 | 0.01 |
| −3 | 1/8 = 0.125 | 1/27 = 0.0370 | 0.001 |
| −4 | 1/16 = 0.0625 | 1/81 = 0.0123 | 0.0001 |
Worked Examples
A whole number exponent: 34
3 × 3 = 9, × 3 = 27, × 3 = 81. So 34 = 81.
A negative exponent: 2−3
Work out the positive power first, 23 = 8, then take the reciprocal: 1/8 = 0.125. A negative exponent never makes the answer negative, it makes it small.
A fractional exponent: 272/3
The bottom of the fraction is the root and the top is the power. The cube root of 27 is 3, and 32 = 9. So 272/3 = 9. Taking the root first keeps the numbers small.
Growth over time: 1.0510
1.0510 = 1.62889, so $1,000 growing at 5% a year for 10 years becomes $1,628.89. The compound interest calculator runs this for any rate and term.
Zero, Negative Bases and Other Edge Cases
- 00. The calculator returns 1, the value used in algebra, combinatorics and most programming languages. In calculus it is treated as an indeterminate form, so check which convention your course uses.
- 0 to a negative power. 0−2 = 1/02 = 1/0, which is undefined. The calculator says so rather than printing infinity.
- Negative bases. (−2)4 = 16 and (−2)5 = −32: an even power makes the result positive, an odd power keeps it negative.
- Negative base, decimal exponent. (−8)0.5 would be the square root of −8, which is not a real number. For a real odd root such as the cube root of −8 = −2, use the nth root option of the square root calculator.
- Very large results. The calculator can hold numbers up to about 1.8 × 10308. 21023 (about 8.99 × 10307) fits, but 21024 does not.
Mistakes to avoid
- −32 is not (−3)2. Without brackets the exponent applies first, so −32 = −9, while (−3)2 = 9. The base box on this calculator always means the bracketed version.
- 23 is not 2 × 3. It is 2 × 2 × 2 = 8, not 6.
- Powers do not spread over addition. (2 + 3)2 = 25, not 22 + 32 = 13.
- Add exponents when multiplying, multiply them when nesting. 23 × 24 = 27 = 128, but (23)4 = 212 = 4,096.
Method and sources. Results use the JavaScript power function (IEEE 754 double precision), shown to 12 significant digits; values of 1015 or more, and below 0.0001, switch to scientific notation. Every value in the tables and examples above was computed exactly and checked. The limits of about 1.8 × 10308 and 5 × 10−324 are the largest and smallest positive double precision numbers.
Laws of Exponents
X^a × X^b = X^(a+b)
Product Rule
2^3 × 2^4 = 2^7 = 128
X^a / X^b = X^(a−b)
Quotient Rule
2^5 / 2^3 = 2^2 = 4
(X^a)^b = X^(a×b)
Power Rule
(2^3)^4 = 2^12 = 4,096
X^0 = 1
Zero Exponent
5^0 = 1, 1000^0 = 1
X^(−n) = 1/X^n
Negative Exponent
2^(−3) = 1/8 = 0.125
X^(1/n) = ⁿ√X
Fractional Exponent
8^(1/3) = ∛8 = 2
(XY)^n = X^n × Y^n
Product Base Rule
(2×3)^2 = 4×9 = 36
(X/Y)^n = X^n / Y^n
Quotient Base Rule
(4/2)^3 = 64/8 = 8
Exponent Questions
An exponent (also called a power) indicates repeated multiplication. In 2^5, the base is 2 and the exponent is 5, meaning 2×2×2×2×2 = 32. Exponents are written as superscripts: 2&sup5; or as 2^5 in text notation. The result is called a power. Exponentiation is the mathematical operation of raising a base to a power.
Any non-zero number raised to the power of 0 equals 1. This is a mathematical definition, not derived from repeated multiplication. 5^0 = 1. 1000^0 = 1. (−7)^0 = 1. The expression 0^0 is considered indeterminate and is typically left undefined, though in some contexts (combinatorics, set theory) it is taken to equal 1 for convenience.
A negative exponent means take the reciprocal. X^(−n) = 1 / X^n. Examples: 2^(−1) = 1/2 = 0.5. 2^(−3) = 1/8 = 0.125. 10^(−2) = 1/100 = 0.01. Negative exponents are used in scientific notation for very small numbers: 3 × 10^−9 = 3 nanometers = 0.000000003.
Fractional exponents represent roots. X^(1/2) = square root of X. X^(1/3) = cube root of X. X^(m/n) = nth root of X raised to the mth power. Example: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4. This connects exponents and roots: x^(1/n) = ⁿ√x. So 9^0.5 = 9^(1/2) = √9 = 3.
The main laws: Product rule: X^a × X^b = X^(a+b). Quotient rule: X^a / X^b = X^(a−b). Power rule: (X^a)^b = X^(ab). Zero exponent: X^0 = 1. Negative exponent: X^(−n) = 1/X^n. Product base: (XY)^n = X^n × Y^n. These rules apply when bases are the same (for product and quotient rules). They allow complex exponential expressions to be simplified.
Exponents grow incredibly fast. 2^10 = 1,024. 2^20 ≈ 1 million. 2^30 ≈ 1 billion. 2^32 = 4,294,967,296 (the number of values a 32-bit integer can hold). 2^64 ≈ 1.8 × 10^19. 2^100 ≈ 1.27 × 10^30. A googol = 10^100. This explosive growth is why compound interest, viral spread, and computer storage capacity follow exponential patterns.
2^32 = 4,294,967,296 (about 4.3 billion). It is the number of different values a 32-bit unsigned integer can hold (0 to 4,294,967,295), which is why 32-bit systems could address only about 4 GB of RAM (2^32 bytes). Similarly, 2^64 ≈ 18.4 quintillion, which is the limit of 64-bit systems. IPv4 has 2^32 addresses; IPv6 has 2^128 addresses.
Strictly: in 2^5, the "exponent" is 5 (the number that indicates how many times to multiply). The "power" is the result: 32. However, in everyday use, these terms are often used interchangeably. "Raise 2 to the power of 5" and "2 to the exponent 5" mean the same thing. "Power of 2" typically means a number of the form 2^n: 1, 2, 4, 8, 16, 32, 64...
Scientific notation expresses numbers as a×10^b, where 1 ≤ |a| < 10. Examples: 299,792,458 (speed of light in m/s) = 2.998 × 10^8. 0.000000001 (1 nanometer) = 1 × 10^−9. The exponent of 10 shows how many places to move the decimal point: positive exponent = large number, negative exponent = small number. It makes very large and very small numbers manageable.
0^0 is mathematically indeterminate (no universally agreed value) because two rules conflict: any number to the 0 = 1, but 0 to any positive power = 0. In practice: in combinatorics and set theory, 0^0 = 1 is used (for example, the number of functions from an empty set to an empty set is 1). In calculus, limits involving 0^0 depend on how you approach it, giving different answers. Calculators often return 1 or an error for this case.
Multiply the negative number by itself as usual and watch the sign. An even exponent gives a positive result and an odd exponent gives a negative one: (−2)4 = 16 and (−2)5 = −32. Brackets matter, because −24 means −(24) = −16. A negative base with a decimal exponent such as 0.5 has no real answer.
Use the caret or the POWER function. =2^10 and =POWER(2,10) both return 1,024. For a root, use a fractional exponent: =27^(1/3) returns 3. Put negative bases in brackets, as in =(−2)^3, which returns −8.