Exponent Calculator
Raise any number to any power, including negative and fractional powers, with every step shown. You can also find a missing exponent, simplify roots like √72 = 6√2, and work out logs and antilogs in any base.
Any box takes whole numbers, decimals, fractions (2/3), negatives, scientific notation (1.5e3), e and π.
2x = 1024
Scientific: 1.024 × 10³. Engineering keeps the power a multiple of 3 (12.5 × 10³). E-notation: 1.024E3.
How your answer is worked out
Your numbers, step by step: read the power, work it out, then check it with logs.
- 1
Base and power
base 2power 102^10
- 2
Multiply it out
2 × 2 × … × 210 times= 1,024
- 3
Count the digits
4 digitsshort enough towrite in full
- 4
Check it
log of answer÷ log 2 = 10back to 10
Read more: the formula, worked examples, exponent and log rules
An exponent says how many times to multiply a number (the base) by itself: 210 is ten 2s multiplied together. A negative exponent flips the answer into 1 over it, and a fractional exponent is a root. Roots and logs are the two ways to undo a power: a root finds the base, and a log finds the exponent.
The formula
b^n = b × b × … × b (n copies of b)
b^−n = 1 ÷ b^n
b^(p/q) = (q-th root of b)^p
q-th root of x = x^(1/q)
log base b of x = the power y where b^y = x = ln x ÷ ln b
antilog base b of y = b^y
if b^x = y then x = log y ÷ log b; if x^n = y then x = y^(1/n)
Worked example
Power (the default): 210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Doubling step by step gives 2, 4, 8, 16, 32, 64, 128, 256, 512 and 1,024, a 4-digit number. Check: log₁₀ 1,024 = 3.010299957 and log₁₀ 2 = 0.3010299957, and 3.010299957 ÷ 0.3010299957 = 10.
Find the exponent: 2x = 1,024, so x = log₂ 1,024. Since 210 = 1,024 exactly, x = 10.
Root: √72. The prime factors are 72 = 23 × 32 = 22 × 32 × 2. One pair of 2s and one pair of 3s come out as 2 × 3 = 6, leaving 2 inside, so √72 = 6√2 ≈ 8.485281374. Check: 6√2 squared is 36 × 2 = 72.
Log: log₁₀ 50 = ln 50 ÷ ln 10 = 3.912023005 ÷ 2.302585093 ≈ 1.698970004. That makes sense: 101 = 10 and 102 = 100, and 50 sits between them.
Antilog: antilog₁₀ 2.5 = 102.5 = 102 × 100.5 = 100 × √10 = 100 × 3.16227766 ≈ 316.227766.
Exponent rules
| Rule | In symbols | Example |
|---|---|---|
| Product | am × an = am+n | 23 × 22 = 25 = 32 |
| Quotient | am ÷ an = am−n | 25 ÷ 22 = 23 = 8 |
| Power of a power | (am)n = am×n | (23)2 = 26 = 64 |
| Power of a product | (ab)n = anbn | (2 × 3)2 = 4 × 9 = 36 |
| Zero power | a0 = 1 | 70 = 1 |
| Negative power | a−n = 1 ÷ an | 2−3 = 1/8 |
| Fractional power | ap/q = (q√a)p | 82/3 = 22 = 4 |
Log rules
- log(xy) = log x + log y, and log(x ÷ y) = log x − log y.
- log(xk) = k × log x. This is why logs can find a missing exponent.
- Change of base: logb x = ln x ÷ ln b = log₁₀ x ÷ log₁₀ b.
- logb b = 1, logb 1 = 0, and logs of 0 or negative numbers have no real answer.
Handy powers and logs
| n | 2n | 10n | log₁₀ n | ln n |
|---|---|---|---|---|
| 1 | 2 | 10 | 0 | 0 |
| 2 | 4 | 100 | 0.30103 | 0.693147 |
| 3 | 8 | 1,000 | 0.477121 | 1.098612 |
| 5 | 32 | 100,000 | 0.69897 | 1.609438 |
| 8 | 256 | 100,000,000 | 0.90309 | 2.079442 |
| 10 | 1,024 | 10,000,000,000 | 1 | 2.302585 |
Negative bases and complex answers
A negative base works with whole-number powers: (−2)3 = −8 and (−2)4 = 16. With a fractional power, an odd root is fine, so (−8)1/3 = −2 (the real cube root). An even root of a negative number is not a real number: √−72 = 6i√2, where i is the square root of −1. For other cases the calculator gives the principal complex value, |b|e × (cos πe + i sin πe).
How big answers stay exact
Whole-number powers are worked out exactly with whole numbers of any size, up to 100,000 digits shown in Advanced. Beyond that the calculator keeps the first 30 digits by repeated squaring and finds the last 20 digits exactly, so for 71,000,000 you still see how many digits it has (845,099), how it starts (1096514191…) and how it ends (…280600000001). Roots and surds are worked out to about 38 digits, and other answers to about 15.
Common questions
How do you work out a negative exponent?
A negative exponent means 1 divided by the same power with a positive exponent. So 2^−3 = 1 ÷ 2^3 = 1 ÷ 8 = 0.125. The minus sign in the exponent never makes the answer negative; only a negative base can do that, like (−2)^3 = −8.
What does a fractional exponent like 2/3 mean?
The bottom of the fraction is a root and the top is a power. So 8^(2/3) means the cube root of 8, squared: ∛8 = 2 and 2^2 = 4. When the root is not a whole number the answer is a surd, for example 2^(3/2) = √8 = 2√2 ≈ 2.828427125.
How do I solve for an exponent?
Take the log of both sides. If b^x = y then x = log y ÷ log b, using any log as long as you use the same one top and bottom. For 2^x = 1,024 the answer is exactly 10. For 10^x = 50 it is ln 50 ÷ ln 10 ≈ 1.698970004.
How do I simplify a square root like √72?
Split the number into prime factors and take out every pair. 72 = 2^3 × 3^2, so one pair of 2s and one pair of 3s come out as 2 × 3 = 6, leaving a single 2 inside. That gives √72 = 6√2 ≈ 8.485281374. For a cube root you take out groups of three.
What is an antilog?
The antilog undoes a log: it raises the base to the number. The common antilog of 2.5 is 10^2.5 ≈ 316.227766, and the natural antilog of 1 is e^1 ≈ 2.718281828. If log₁₀ x = 2.5, then x = 316.227766.
Why is any number to the power 0 equal to 1?
Because dividing powers subtracts exponents: b^1 ÷ b^1 = b^(1−1) = b^0, and anything divided by itself is 1. This works for every base except 0. 0^0 is usually taken as 1 in algebra and computing, though some books leave it undefined.
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