Negative Exponent Calculator with Reciprocal Steps
Work out a number raised to a negative power such as 2^-3. See the reciprocal rewrite, the exact fraction 1/8 and the decimal 0.125, free in your browser.
Raise the base to an exponent: base^exponent. The exponent may be negative, fractional such as 1/3, or zero.
2^-3 =
0.125
- Reciprocal:
- 2^-3 = 1 / 2^3
- Expanded:
- 1 / (2 x 2 x 2) = 0.125
- Fraction:
- 1/8
Negative Exponent Calculator
This page opens on a worked example: 2 raised to the power -3. A negative exponent does not make the answer negative. It means "take the reciprocal", so 2^-3 is the same as 1 divided by 2^3, which is 1/8 or 0.125. The calculator shows that rewrite, the exact fraction and the decimal, and you can type any other base and exponent to see the same steps.
What a negative exponent means
Positive exponents count how many times a base is multiplied. Negative exponents count how many times you divide by it. Each step down in the exponent divides the result by the base, which is why the pattern 2^3 = 8, 2^2 = 4, 2^1 = 2, 2^0 = 1 carries on naturally to 2^-1 = 1/2, 2^-2 = 1/4 and 2^-3 = 1/8.
The second form is useful for fractions: a negative exponent flips the fraction. For example, (2/3)^-2 equals (3/2)^2, which is 9/4.
How to calculate a negative exponent
Steps
- Leave the page on Power mode. The Base is 2 and the Exponent is -3 to start with. Replace them with your own numbers.
- Read the Reciprocal line, which rewrites the power with a positive exponent: 2^-3 = 1 / 2^3.
- For whole exponents up to 12 in size, the Expanded line shows the product, such as 1 / (2 x 2 x 2) = 0.125.
- When the base and exponent are whole numbers, the Fraction line gives the exact fraction, such as 1/8, and the answer shows the decimal to 10 significant digits.
Worked examples
Example 1: 3 to the power -4
Rewrite as 1 / 3^4, which is 1 / (3 x 3 x 3 x 3) = 1/81. As a decimal that is 0.01234567901.
Example 2: 10 to the power -4
10^-4 = 1 / (10 x 10 x 10 x 10) = 1/10000 = 0.0001. Negative powers of ten are how small numbers are written in scientific notation, so 10^-4 is a ten-thousandth.
Example 3: a negative base
(-2)^-3 = 1 / (-2)^3 = 1 / -8 = -1/8 = -0.125. The sign follows the base: an odd exponent keeps the negative sign and an even one removes it, so (-2)^-4 is positive 1/16.
Rules for exponents
- Product rule: x^a times x^b equals x^(a+b), so 2^5 times 2^-3 equals 2^2, which is 4.
- Quotient rule: x^a divided by x^b equals x^(a-b).
- Power of a power: (x^a)^b equals x^(a times b).
- Zero exponent: x^0 is 1 for any non-zero x.
- Zero base: 0 to a negative power is undefined because it divides by zero, and the calculator reports that as an error. 0^0 is shown as 1 with a note that it is indeterminate.
Precision and limits
Whole-number bases with whole exponents are computed exactly. Reciprocals that have more than 60 digits in the denominator, such as 2^-300, appear in scientific notation, for example 4.909093465e-91. Other results are shown to 12 significant digits. A true result too small for a double shows as 0 with a note that it rounds to 0.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
The general Exponent Calculator also finds the exponent for a given result and the n-th root of a number. The Percentage Calculator is useful for scaling values, and the Pascal's Triangle Calculator shows the coefficients behind expanded powers.
Frequently asked questions
What does a negative exponent mean?
It means take the reciprocal of the power with the matching positive exponent. So 2^-3 is 1 / 2^3, which is 1/8 or 0.125. The exponent being negative does not make the answer negative.
Can the answer to a negative exponent be negative?
Only when the base is negative and the exponent is an odd whole number. (-2)^-3 is 1 / (-2)^3 = -1/8, while (-2)^-4 is positive 1/16. With a positive base the answer is always positive.
How do I write a negative exponent as a fraction?
Put 1 over the base raised to the positive exponent. For whole numbers the calculator shows the exact fraction, such as 3^-4 = 1/81, and the decimal to 10 significant digits, 0.01234567901. When the denominator would exceed 60 digits it shows scientific notation instead.
What is a negative exponent on a fraction?
It flips the fraction and makes the exponent positive: (2/3)^-2 equals (3/2)^2, which is 9/4. You can enter the base as a decimal, for example 0.5, to see this with a calculator.
Why can I not raise 0 to a negative power?
0^-n means 1 / 0^n, and 0^n is 0, so the calculation divides by zero. The calculator reports an error. Zero to the power 0 is a separate case that is shown as 1 with a note that it is indeterminate.
How do negative exponents relate to scientific notation?
Negative powers of ten are how small numbers are written: 10^-4 is 0.0001 and 10^-9 is 0.000000001. A result such as 4.909093465e-91 means 4.909093465 times 10^-91.