Exponent calculator, powers, roots and big numbers.

An exponent calculator works out x to the power y: the number x multiplied by itself y times. This one handles negative, decimal and fractional exponents, roots written as exponents, e to the x and 10 to the x. Whole number powers are exact to the last digit, with the digit count, and every small case shows its steps.

How to use the exponent calculator

1. Choose what to work out: x to the power y, e to the x, 10 to the x or Nth root.
2. Type the Base x and the Exponent y. Decimals such as 0.5 and fractions such as 1/3 are fine.
3. Read the big answer, the scientific notation and, for whole numbers, the digit count. The steps and the formula sit below.
4. Use Copy result for the line, Copy all digits for a very long number, or Copy link to share the sum.
5. Under the answer, Powers of the base lists x⁰ up to the power you choose, and the rules table checks the product, quotient and power rules for any m and n.

The rules of exponents

• Product: xᵐ × xⁿ = x^(m+n). Add the exponents when the base is the same. 3² × 3⁴ = 9 × 81 = 729 = 3⁶.
• Quotient: xᵐ ÷ xⁿ = x^(m−n). Subtract them.
• Power of a power: (xᵐ)ⁿ = x^(m×n). Multiply them.
• Zero: x⁰ = 1 for any x that is not 0.
• Negative: x^(−n) = 1 ÷ xⁿ. 5⁻² = 1 ÷ 25 = 0.04.
• Fraction: x^(p/q) is the q-th root of x, raised to p. A square root is the exponent 1/2.

Worked examples

2 to the 10. 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024, or 1.024 × 10³ in scientific notation.

A fractional exponent. 8^(2/3) is the cube root of 8 squared. The cube root of 8 is exactly 2, and 2² = 4. The calculator spots perfect roots like this and gives the exact answer. For 2^(1/2) there is no exact fraction, so it gives the decimal 1.41421356237.

A negative base. (−8)^(1/3) is −2, because a cube root of a negative number is real. (−4)^(1/2) has no real answer, so the calculator says so and shows the complex value 0 + 2i.

A big power. 2^100 is 1,267,650,600,228,229,401,496,703,205,376, a number with 31 digits. A normal calculator would round it to 1.26765E+30. Here every digit is kept, and the copy button copies them all. The most it writes out in full is about 200,000 digits; past that it gives scientific notation and says it is rounded.

Powers of ten. 10^(−3) is 1 ÷ 1,000 = 0.001, and 10^6 is 1,000,000. These are the steps behind scientific notation: a number like 4,500 is 4.5 × 10³.

Roots are exponents

The square root of x is x^(1/2) and the cube root is x^(1/3). Pick Nth root, type the number and the root, and the calculator writes the root as an exponent for you. The cube root of 64 is 64^(1/3) = 4, because 4 × 4 × 4 = 64. Cube roots of negative numbers are real, so the cube root of −27 is −3.

Common powers to know

Watch the minus sign

The base box takes the whole number, sign included. Typing −3 as the base and 2 as the exponent gives (−3)² = 9. Written on paper without brackets, −3² means −(3²) = −9. If you want that, type 3 as the base and put the minus on the answer yourself.

e to the x

e is about 2.718281828459045. e^x is the function that grows at a rate equal to its own size, and it is the opposite of the natural log. e^1 is 2.71828182846, and e^(−1) is 0.367879441171. Choose e to the x and type x.

Common problems

• 0 to the power 0. It is shown as 1, the usual convention. Some areas of maths leave it undefined, and the page says so.
• 0 to a negative power. It means dividing by 0, so there is no value.
• Decimal exponents. 2.5 is read as exactly 5/2, so 4^2.5 is the square root of 4, to the power 5, which is 32.
• Rounded decimals. When the answer is not a whole number or a simple fraction it is rounded to 12 significant digits. Floating point numbers cannot hold more than about 15 to 17.

For longer sums with brackets use the scientific calculator, to plot y = xⁿ try the graphing calculator, and for percentages of a result see the percentage calculator.