Understanding exponent results
The sign and size of the exponent determine the character of the result. The table below summarizes the cases for a base greater than 1.
| Exponent | Result (base > 1) | Example |
|---|---|---|
| Positive integer | Repeated multiplication; grows rapidly | 2^10 = 1024 |
| Zero | Always 1 for any nonzero base | 7^0 = 1 |
| Negative | Reciprocal of the positive power; between 0 and 1 | 2^-3 = 0.125 |
| Fraction 1/n | The n-th root | 9^0.5 = 3 |
| Fractional, negative base | Generally not a real number | (-8)^0.5 undefined in the reals |
- Exponentiation is not commutative: 2^10 = 1024 but 10^2 = 100.
- Repeated exponentiation is conventionally evaluated right-to-left: 2^3^2 means 2^(3^2) = 2^9 = 512, not (2^3)^2 = 64.
- Results beyond about 10^15 lose integer precision in double-precision floating point and are shown in scientific notation.
- 0^0 is treated as 1 by most computing environments (including this calculator), though it is left undefined in some mathematical contexts.
What is exponentiation?
Exponentiation writes repeated multiplication compactly: b^n means the base b multiplied by itself n times when n is a positive whole number. So 2^10 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1024. The base is the number being multiplied; the exponent (or power, or index) counts the multiplications.
The definition extends consistently beyond positive whole numbers. Any nonzero base to the power 0 equals 1. A negative exponent gives the reciprocal: b^(-n) = 1 / b^n, so 2^(-3) = 1/8 = 0.125. A fractional exponent gives a root: b^(1/2) is the square root of b and b^(1/3) the cube root, so 9^0.5 = 3. These extensions are forced by requiring the law b^(m+n) = b^m x b^n to keep holding.
Fractional powers of negative bases are generally not real numbers — for example, (-8)^0.5 would be the square root of a negative number — so this calculator returns no result in those cases. Very large results are displayed in scientific notation, the standard m x 10^k format used throughout science and engineering.
How to use this exponent calculator
- Enter the base — the number to be raised to a power. It may be negative or a decimal.
- Enter the exponent. Positive, negative, zero and fractional exponents are all supported.
- Read the result. Values larger than about 10^15 in magnitude are shown in scientific notation.
- Read the scientific-notation form (m x 10^k, displayed as an e-notation string such as 1.024e+3), which is always provided for reference.
The laws of exponents
The core rules: multiplying powers of the same base adds the exponents, b^m x b^n = b^(m+n); dividing subtracts them; raising a power to a power multiplies them, (b^m)^n = b^(mn); any nonzero base to the 0 equals 1; a negative exponent means a reciprocal; and a fractional exponent 1/n means the n-th root.
Worked example: 2^10. Multiply ten factors of 2: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 — so 2^10 = 1024. In scientific notation, 1024 = 1.024 x 10^3, which this calculator displays as 1.024e+3.
Negative exponent example: 2^(-3) = 1 / 2^3 = 1/8 = 0.125. Fractional exponent example: 9^0.5 = sqrt(9) = 3, and 8^(1/3) = 2 because 2^3 = 8.
Common mistakes
- Multiplying the base by the exponent: 2^10 is 1024, not 20.
- Treating a negative exponent as producing a negative number — b^(-n) is the reciprocal 1/b^n, which is positive for positive b.
- Confusing -2^4 with (-2)^4: by convention exponentiation binds before negation, so -2^4 = -16 while (-2)^4 = 16.
- Adding exponents when the bases differ — the rule b^m x b^n = b^(m+n) applies only to identical bases.
- Expecting a real result for a fractional power of a negative base, such as (-8)^0.5, which is not a real number.
Câu hỏi thường gặp
What is 2 to the power of 10?
2^10 = 1024. It is 2 multiplied by itself ten times, and in scientific notation it is written 1.024 x 10^3. Powers of 2 are fundamental in computing: 2^10 bytes is one kibibyte.
What does a negative exponent mean?
A negative exponent means the reciprocal of the corresponding positive power: b^(-n) = 1 / b^n. For example, 2^(-3) = 1 / 2^3 = 1/8 = 0.125. The result of a negative exponent on a positive base is always positive — it is small, not negative.
What does a fractional exponent mean?
A fractional exponent denotes a root: b^(1/n) is the n-th root of b, and b^(m/n) is the n-th root of b^m. For example, 9^0.5 = sqrt(9) = 3 and 8^(2/3) = (cube root of 8)^2 = 2^2 = 4. Fractional powers of negative bases are generally not real numbers.
Why does any number to the power 0 equal 1?
It follows from the law of exponents b^m / b^n = b^(m-n). Setting m = n gives b^m / b^m = b^0, and any nonzero number divided by itself is 1, so b^0 = 1. The expression 0^0 is a special case: most computing systems evaluate it as 1, but some mathematical contexts leave it undefined.
What is scientific notation?
Scientific notation writes a number as m x 10^k, where the mantissa m has one nonzero digit before the decimal point and k is an integer power of ten. For example, 1024 = 1.024 x 10^3 and 0.00052 = 5.2 x 10^-4. It keeps very large and very small numbers readable and makes orders of magnitude explicit; calculators often display it in e-notation, e.g. 1.024e+3.
Tài liệu tham khảo
- Weisstein, Eric W. "Exponent Laws." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- ISO 80000-2:2019. Quantities and units — Part 2: Mathematics (notation for powers and roots).
- Stewart J. Precalculus: Mathematics for Calculus. Cengage (exponents, radicals and scientific notation).