Exponent Calculator
Calculate any base raised to any power, including negative exponents.
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About
Exponent Calculator
Enter values into any two of the three fields above and the calculator solves for the third, with the working shown underneath.
What is an exponent?
Exponentiation is a mathematical operation written as an, involving the base a and the exponent n. Where n is a positive integer, it corresponds to multiplying the base by itself n times.
an = a × a × a × … × a, n times
where:
- a is the base, the number being multiplied
- n is the exponent, the number of times it appears in the product
So 103 is 10 × 10 × 10, which is 1,000. Read aloud it is "ten to the third" or "ten cubed". The second power has its own name too: 52 is "five squared", from the area of a square with sides of 5.
The calculator accepts negative bases but does not compute imaginary numbers. It does not accept fractions written as fractions, though it handles fractional exponents perfectly well when they are entered in decimal form: 5/7 goes in as 0.714286.
The exponent laws
Multiplying powers with the same base
When powers sharing a base are multiplied, the exponents are added.
an × am = a(n+m)
22 × 24 = 4 × 16 = 64, and 2(2+4) = 26 = 64. The reason is visible if you write it out: two 2s multiplied by four 2s is six 2s.
Negative exponents
A negative sign on the exponent is removed by taking the reciprocal of the base and raising it to the positive power.
a(-n) = 1 / an
2(-3) is 1 ÷ 2 ÷ 2 ÷ 2, which is 1/8. Equally, 2(-3) = 1 / 23 = 1/8. A negative exponent means division rather than a negative answer, which is the most common misreading of the notation.
Dividing powers with the same base
When powers sharing a base are divided, the exponents are subtracted.
am / an = a(m-n)
22 / 24 = 4/16 = 1/4, and 2(2-4) = 2-2 = 1/22 = 1/4.
A power raised to a power
When a power is raised to another exponent, the exponents multiply.
(am)n = a(m×n)
(22)4 = 44 = 256, and 2(2×4) = 28 = 256.
A product raised to a power
When multiplied bases are raised to an exponent, the exponent distributes across both.
(a × b)n = an × bn
(2 × 4)2 = 82 = 64, and 22 × 42 = 4 × 16 = 64.
A quotient raised to a power
Division behaves the same way.
(a / b)n = an / bn
(2/5)2 = 2/5 × 2/5 = 4/25, and 22 / 52 = 4/25.
Note that this distribution works over multiplication and division only. It does not work over addition: (a + b)2 is not a2 + b2, a mistake common enough to have a nickname among teachers.
Exponents of 1 and 0
An exponent of 1 leaves the base unchanged: a1 = a.
An exponent of 0 gives 1 for any base, though some debate surrounds whether 00 is 1 or undefined. For many applications defining it as 1 is convenient, and this calculator does.
a0 = 1
The reason follows from the first law. If an × am = a(n+m), then an × a0 = a(n+0) = an. The only value that leaves an unchanged under multiplication is 1, so a0 has to be 1 for the law to hold.
Fractional exponents and roots
When an exponent is a fraction with 1 on top, the calculation is a root.
a(1/n) = n√a
So 8(1/3) is the cube root of 8, which is 2. Where the numerator is not 1, both this rule and the power-of-a-power rule apply together.
Take 3(5/7). That is (3(1/7))5, the seventh root of 3 raised to the fifth power, which is 1.175, or about 2.19. The calculator handles fractional exponents, but they have to be entered in decimal form.
This is why roots and powers are the same operation seen from different sides. A square root is a power of 0.5, and the root calculator and this one are solving the same equation for different unknowns.
Negative bases
Exponents with negative bases follow much the same rules as positive ones. Raised to positive integers, the results match their positive counterparts in magnitude and differ only in sign. With an even exponent the result is positive whichever sign the base has. With an odd exponent the magnitude is the same and the sign is negative.
Fractional exponents with negative bases follow the same rules in principle, but they involve imaginary numbers, since no real number can be an even root of a negative number. This calculator cannot compute imaginary results, and any input producing one returns "not a number".
For reference, the numerical solution is the same as for a positive base, with the answer denoted as imaginary: -2(1/2) is √-2, which is 1.414i.
A note on how it is written
Notation causes more errors here than the arithmetic does. Under standard order of operations, exponentiation binds more tightly than the negative sign, so -22 means -(22), which is -4. To square negative two you have to write (-2)2, which is 4. Spreadsheets and programming languages differ on this, and it is worth checking which convention a tool uses before trusting the output.
Exponentiation is also right-associative: 232 means 2(32) = 29 = 512, not (23)2 = 64.
Solving for the exponent
Fill in the base and the result and the calculator works backwards to the exponent using logarithms.
n = log(result) / log(base)
Given 10n = 1,000, that is log(1000) / log(10) = 3. This is the operation logarithms exist for: an exponent asks what you get from repeated multiplication, and a logarithm asks how many multiplications got you there.
It has conditions. The base must be positive and not 1, since the logarithm of zero or a negative number is undefined and every power of 1 is 1. The result must be positive too, because a positive base raised to any real power is always positive.
Where exponents turn up
Compound interest is exponentiation: a balance multiplied by (1 + r) once per period is multiplied by (1 + r)n over n periods. Population growth, radioactive decay and the cooling of a hot object all follow the same shape, with decay simply using a base under 1.
Scientific notation exists because exponents make unwieldy numbers manageable. The speed of light is 3 × 108 metres per second rather than 300,000,000, and a hydrogen atom is about 1.06 × 10-10 metres across.
Computing runs on powers of 2, which is why storage comes in sizes like 256 and 1,024. Algorithm cost is described the same way: an O(2n) algorithm becomes unusable at values of n where an O(n2) one is still comfortable.
The constant e, roughly 2.718, is the base that makes continuous growth work out cleanly, which is why it appears throughout calculus, statistics and finance. The button above fills it in for you.
The thing worth carrying away is how fast exponential growth moves. Doubling a grain of rice on each of a chessboard's 64 squares ends at 263 grains on the final square alone, which is more rice than has ever been grown. Intuition consistently underestimates this, which is exactly why the arithmetic is worth doing rather than guessing.
Common questions
Frequently asked questions
The exponent is the small raised number in a power. In a to the n, a is the base and n is the exponent, and for a positive integer n it means multiplying the base by itself n times. So 10 to the third is 10 x 10 x 10, which is 1,000.
1, for any non-zero base. It follows from the rule that exponents add when powers multiply: a to the n times a to the 0 must equal a to the n, and the only value leaving it unchanged is 1. Whether 0 to the 0 is 1 or undefined is debated; this calculator returns 1.
Division rather than a negative answer. A negative exponent means the reciprocal of the positive power, so 2 to the minus 3 is 1 divided by 2 cubed, which is 1/8. The result stays positive when the base is positive.
A fractional exponent is a root. A power of 1/n is the nth root, so 8 to the 1/3 is the cube root of 8, which is 2. Where the numerator is not 1, take the root then raise to the numerator: 3 to the 5/7 is the seventh root of 3 raised to the fifth, about 2.19. Enter fractional exponents in decimal form.
Yes for integer exponents. An even exponent gives a positive result and an odd one keeps the sign negative. A fractional exponent on a negative base produces an imaginary number, which this calculator cannot compute and reports as not a number.
Use logarithms: n equals log of the result divided by log of the base. For 10 to the n equals 1,000, that is log(1000) divided by log(10), which is 3. Enter the base and the result above and the calculator does it for you.
Because exponentiation binds more tightly than the negative sign, so -2 squared reads as the negative of 2 squared, which is -4. To square negative two you have to write it as (-2) squared, giving 4. Tools differ on this, so check the convention before trusting the output.
They are the same operation from opposite sides. A root is a fractional exponent: the square root is a power of 0.5 and the cube root a power of one third. Exponentiation asks what repeated multiplication produces; a root asks what was multiplied.
e is roughly 2.718, the base that makes continuous growth work out cleanly, which is why it runs through calculus, statistics, and compound interest. Use it whenever growth or decay is continuous rather than stepped. The button above fills it in.
Because the inputs have no real answer. The usual causes are a negative base with a fractional exponent, which needs imaginary numbers, or solving for an exponent with a base or result that is zero or negative, where the logarithm is undefined.