Exponent Calculator

What does the exponent calculator do?

It has three jobs. The power tab finds a power of a number: the base can be a whole number, a decimal or a fraction, and the exponent a whole number or a fraction. The rules tab simplifies powers of the same base and shows which rule is used. The scientific notation tab writes a number as a × 10ⁿ or expands a number given in that form.

What is an exponent?

aⁿ means a multiplied by itself n times: 2⁵ = 2·2·2·2·2 = 32. For whole number exponents the result is given in full with every digit (all 31 digits of 2¹⁰⁰ are shown) together with its scientific notation.

• Zero exponent: a⁰ = 1 for every non-zero number.
• Negative exponent: a⁻ⁿ = 1/aⁿ. Example: 2⁻³ = 1/8.
• Fractional exponent: a^(r/s) is the r-th power of the s-th root of a. Example: 8^(2/3) = (∛8)² = 2² = 4.
• An even power of a negative base is positive and an odd power is negative.

Exponent rules

• Product: aᵐ · aⁿ = aᵐ⁺ⁿ
• Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• Power of a power: (aᵐ)ⁿ = aᵐ·ⁿ
• Power of a product: (a · b)ⁿ = aⁿ · bⁿ
• Power of a quotient: (a / b)ⁿ = aⁿ / bⁿ

The base can be a letter (x) or a number. With a number the value of the result is worked out as well. Example: 2⁵ ÷ 2⁸ = 2⁻³ = 1/8.

Scientific notation

Very large and very small numbers are written as a × 10ⁿ, where the coefficient a is between 1 and 10. In 0.00045 the decimal point moves 4 places to the right, giving 4.5 × 10⁻⁴. In 123000 it moves 5 places to the left: 1.23 × 10⁵. To go the other way type 4.5×10^-4 or 4.5e-4. The tool also gives engineering notation, where the exponent is a multiple of 3.

Limits

0⁰ is indeterminate and a negative power of zero is undefined, so neither is calculated. An even root of a negative number is not real, so (−4)^(1/2) has no result. An exact result can have at most 3000 digits. For fractional exponents whose root is not exact the result is a decimal approximation. With fractional exponents the rules hold for positive bases only. In scientific notation the digits are kept as typed, without trailing zeros.