Exponent & Root Calculator
A negative exponent is a reciprocal. A fractional one is a root. Both follow from the same rule, and knowing that removes most of the mystery.
9 to the power of 3
729.000000
- 9 to the power of 3
- 729.000000
- Squared
- 81.000000
- Cubed
- 729.000000
- Square root
- 3.000000
- Cube root
- 2.080084
- A negative exponent is a reciprocal: 2 to the power of −3 is one eighth. A fractional exponent is a root, so the power of 0.5 is the square root and 1/3 is the cube root.
- Anything to the power of zero is one, including zero to the power of zero by most conventions. It follows from the rule that dividing powers subtracts exponents.
About exponents and roots
An exponent is repeated multiplication: 9 to the power of 3 is 9 × 9 × 9, or 729. That much is straightforward. What confuses people is what happens when the exponent is not a positive whole number, and the answer is that everything follows from one rule about consistency.
Dividing powers subtracts exponents, so 9³ ÷ 9² is 9¹. Follow that down and 9¹ ÷ 9¹ must be 9⁰, which is 1 — so anything to the power of zero is one. Carry on and 9⁰ ÷ 9¹ is 9⁻¹, which is one ninth. A negative exponent is a reciprocal, not a negative number.
Fractional exponents work the same way from the rule that multiplying powers adds exponents. Something to the power of a half, multiplied by itself, gives the power of one — which is the definition of a square root. So 9 to the power of 0.5 is 3, and to the power of a third is the cube root.
Negative numbers have no real square root, because nothing multiplied by itself gives a negative. They do have a real cube root: minus two cubed is minus eight, so the cube root of minus eight is minus two. That asymmetry catches people out and is why the two are reported separately here.
What it works out
- Any base to any exponent, including negative and fractional
- Squares, cubes, square roots and cube roots
- Negative bases handled correctly
The formula
bⁿ b⁻ⁿ = 1 ÷ bⁿ b^(1/n) = the nth root of b
Nine to the power of three is 9 × 9 × 9 = 729.
The other cases fall out of two rules. Multiplying powers adds exponents; dividing them subtracts. Everything else is a consequence.
Zero: 9² ÷ 9² is 9⁰ by the subtraction rule, and it is obviously 1 by cancellation. So anything to the power of zero is one.
Negative: 9⁰ ÷ 9¹ is 9⁻¹, and it is 1 ÷ 9 by cancellation. So a negative exponent is a reciprocal — 2⁻³ is one eighth, not minus eight.
Fractional: 9^½ × 9^½ is 9¹ by the addition rule, so 9^½ is whatever multiplies by itself to make 9. That is the square root, 3. A power of one third is the cube root.
The exponent is where growth lives. Nine squared is 81; nine cubed is 729. Changing the exponent moves the answer far more than changing the base, which is what "exponential" means when people use it loosely.
- Base
- The number being multiplied by itself.
- Exponent
- How many times. Negative gives a reciprocal, fractional gives a root.
- Root
- The inverse of a power. The square root undoes squaring.
A worked example
Nine to the power of three.
That works out to 729.000000 .
- 9 to the power of 3
- 729.000000
- Squared
- 81.000000
- Cubed
- 729.000000
- Square root
- 3.000000
- Cube root
- 2.080084
Questions
What does a negative exponent mean?
A reciprocal. 2 to the power of −3 is one eighth, not minus eight. It follows from the rule that dividing powers subtracts exponents.
What is a fractional exponent?
A root. The power of 0.5 is the square root, one third is the cube root, and one nth is the nth root. Nine to the power of a half is three.
Why is anything to the power of zero equal to one?
Because 9² ÷ 9² is 9⁰ by the subtraction rule and 1 by cancellation. Both have to be true, so the power of zero is one — for every base except zero itself, where it is a matter of convention.
Can I take the square root of a negative number?
Not a real one. Nothing multiplied by itself gives a negative. Imaginary numbers exist for exactly this and are outside what a calculator like this deals in.
Why does a negative number have a cube root?
Because three negatives multiplied together stay negative. Minus two cubed is minus eight, so the cube root of minus eight is minus two. Odd roots work on negatives and even ones do not.
What is the difference between a power and a multiple?
A multiple adds repeatedly — 9 × 3 is 9 + 9 + 9. A power multiplies repeatedly — 9³ is 9 × 9 × 9. The first gives 27 and the second 729, which is why the distinction matters.