Power, root and logarithm calculator

Zero and negative exponents

Any non-zero number raised to the power 0 equals 1. To see why, look at 5³ ÷ 5³: on one hand it is 125 ÷ 125 = 1, on the other the exponents subtract to give 5⁰. So 5⁰ has to be 1.

A negative exponent means a reciprocal: a⁻ⁿ = 1 ÷ aⁿ. For example 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125 and 10⁻² = 0.01. A negative exponent does not make the result negative; it makes it smaller than 1 (when the base is greater than 1).

Negative powers of ten are used in scientific notation for very small numbers: 3.2 × 10⁻⁴ = 0.00032. In the calculator you get 10⁻⁴ by entering 10 as the base and -4 as the exponent.

A fractional exponent is a root

The exponent 1/n means the nth root, and more generally a^(m/n) = (nth root of a)^m. For example 8^(2/3): the cube root of 8 is 2, and 2² = 4. Likewise 16^0.75 = 16^(3/4): the fourth root of 16 is 2, and 2³ = 8.

A square root is the same as the power 0.5, so √2 = 2^0.5 ≈ 1.414214. The calculator rounds to six decimal places because most roots are irrational numbers that never end.

Be careful with roots of negative numbers. An odd root exists: the cube root of -27 is -3, since (-3)³ = -27, and the calculator's root mode gives that answer. An even root of a negative number (such as √-4) does not exist among the real numbers, and the calculator shows "undefined". In power mode a negative base combined with a fractional exponent (such as -8 and 0.333333) does not give a correct answer, so use root mode for the cube root of a negative number.

The laws of exponents with examples

These rules apply to powers with the same base and let you simplify an expression before calculating anything:

Powers also show up in everyday percentages. If a price rises 10% three years in a row, the factor is 1.1³ = 1.331, a total of +33.1%, not +30%. Each year the rise is calculated on the previous year's price, so growth is a question of powers, not addition.

Roots can often be simplified by taking a perfect square out from under the root sign: √50 = √(25 × 2) = 5√2 ≈ 7.071068.

• Multiplying adds the exponents: 2³ × 2⁴ = 2⁷ = 128.
• Dividing subtracts the exponents: 3⁵ ÷ 3² = 3³ = 27.
• A power of a power multiplies the exponents: (2³)² = 2⁶ = 64.
• A stacked power without brackets is evaluated from the top down: 2^(3²) = 2⁹ = 512, which is not the same as (2³)².
• Power of a product: (2 × 5)³ = 2³ × 5³ = 1000.

Logarithms in practice: how long until money doubles

A logarithm answers the question "to what power?". log₁₀1000 = 3 because 10³ = 1000, and log₂1024 = 10 because 2¹⁰ = 1024. The common logarithm also gives the number of digits: its whole part plus one. log₁₀100000 = 5, and 100000 has 6 digits.

A useful application is growth time. If savings grow 5% a year, we look for n such that 1.05ⁿ = 2. The answer is log₁.₀₅2 ≈ 14.21 years. Check: 1.05¹⁴ ≈ 1.98 and 1.05¹⁵ ≈ 2.08, so the money doubles during the fifteenth year. In the calculator, enter 2 as the number and 1.05 as the base.

The laws of logarithms mirror those of exponents: log(x × y) = log x + log y and log(x ÷ y) = log x - log y. A logarithm is defined only for positive numbers, and the base must be positive and not equal to one; otherwise the calculator shows "undefined".

Common mistakes with powers and roots

Keep rounding in mind too. Because results are shown to six decimal places, a very small number such as 2⁻³⁰ ≈ 0.00000000093 is displayed as zero. In that case write the answer in scientific notation or use a smaller exponent.

Most mistakes come from misreading brackets and signs, or from trying to distribute a power or a root over a sum as if it were multiplication.

• -2² and (-2)² are different: the first is -(2²) = -4, the second is 4. Entering a negative base in the calculator corresponds to the bracketed form.
• The square of a sum is not the sum of squares: (3 + 4)² = 49, but 3² + 4² = 25.
• The root of a sum is not the sum of roots: √(9 + 16) = √25 = 5, but √9 + √16 = 7.
• You cannot swap base and exponent: 2³ = 8, but 3² = 9.
• 0⁰ is a debated case in mathematics; the calculator returns 1 by convention.