How this instrument works
Raising a number to a power means repeated multiplication when that power is a positive whole number, but the operation extends smoothly past that starting point: base^exponent stays fully defined even when the power is negative, fractional, or an ordinary decimal. This sheet accepts any of the three directly in the Exponent field and returns the result to full precision, whatever shape that figure takes.
A whole positive power multiplies the number by itself that many times, so 2^10 is ten 2s multiplied together, landing on 1,024. A negative power flips that number into a denominator instead: 5^−2 equals 1 divided by 5^2, or 1⁄25, which comes out to 0.04. A fractional power asks for a root rather than a further multiplication — 8^(1⁄3) takes the cube root of 8, and since 2 × 2 × 2 = 8, the answer lands on exactly 2.0. Three operations that look unrelated on the surface, one formula covering every one of them underneath.
Other sheets on this site each handle one slice of this same calculation at a time — Power of 2 fixes its input at 2 and only accepts a whole-number power, Power of 10 does the same at 10, and Fractional Exponent restricts what it takes to a ratio m⁄n entered as a separate numerator and denominator. This page keeps both fields open: any starting number, any real power, typed in directly rather than picked from a fixed list, which makes it the general version those narrower sheets each carve a single restricted case out of.
- Enter the number being raised to a power into the Base field — any real number, positive, negative, or a decimal.
- Enter the power itself into the Exponent (any real number) field — whole, negative, or fractional values are all accepted directly.
- Read the answer in the base^exponent field, computed to full precision no matter which kind of power was entered.
- Try a negative Exponent, such as −2, to watch the result flip into a fraction smaller than 1.
- Try a fractional Exponent, such as 1/3 entered as 0.3333333333333333, to see a root computed instead of a repeated multiplication.
Worked example — three exponents, three regimes
Enter Base = 2 and Exponent = 10, and base^exponent returns exactly 1024 — ten 2s multiplied together, 2×2×2×2×2×2×2×2×2×2 = 1024, with no rounding needed since every factor involved is a whole number.
Change the inputs to Base = 5 and Exponent = −2, and the result flips into a fraction: 5^−2 = 1 ⁄ 5^2 = 1 ⁄ 25 = 0.04, the negative sign sending that number to the bottom of a fraction rather than making the answer itself negative. Switch again to Base = 8 and Exponent = 1/3, entered as 0.3333333333333333, and base^exponent returns 2.0 exactly, because 2 cubed is 8 and a one-third power asks for precisely that cube root.
Questions
What does raising a base to an exponent mean when the exponent is not a whole number?
It means a root rather than repeated multiplication — an exponent of 1⁄n asks for the n-th root of the base, so 8^(1/3) takes the cube root of 8 and returns 2.0, since 2 × 2 × 2 = 8. Whole, negative, and fractional powers all reduce to this one general operation, base^exponent, computed the same way no matter which kind is entered.
What happens with a negative exponent?
That number flips into the denominator of a fraction: base^(−n) equals 1 divided by base^n. With Base = 5 and Exponent = −2, that works out to 1 ⁄ 5² = 1 ⁄ 25 = 0.04 — a fraction smaller than 1, not a negative number, since the minus sign marks a reciprocal rather than subtraction.
How does this sheet differ from the Power of 2 and Power of 10 calculators?
Those two fix that starting number ahead of time — always 2, or always 10 — and only accept a whole-number exponent. This sheet leaves the base open to any real number as well, so a query like 5^−2 or 8^(1/3), which neither of those narrower sheets can take, computes here directly, in one field, with nothing preset.
How is this different from the Fractional Exponent calculator?
That sheet takes the exponent apart into a numerator and a denominator entered separately, m and n, and shows the root-then-power steps individually. This page instead accepts it as a single figure typed straight into one field, whole, negative, or fractional alike, and returns base^exponent directly without splitting the fraction into two parts first.
Can the base be negative?
Yes, for a whole-number exponent — (−2)^3 is a real −8. Pairing a negative base with a non-whole exponent, though, has no real result, since it would require an even root of a negative number; this sheet's check flags that combination rather than returning a false answer.
What is base raised to the power 0?
It equals 1 for any nonzero base, since raising a number to the power 0 always returns the multiplicative identity regardless of what that number itself is. This holds whether it is 2, 8, or 5 — only 0 raised to the power 0 sits outside the rule, left undefined rather than settled at 1.