Raise any number to any power, including negative powers (which give a reciprocal) and fraction powers (which give a root).
A whole-number exponent means repeated multiplication of the base.
A negative exponent gives the reciprocal, so 2⁻² is 1 ÷ 4 = 0.25.
A fractional exponent gives a root, so 9^0.5 is the square root of 9.
Any non-zero number raised to the power 0 equals 1.
Power: bⁿ = b × b × … n times
Negative exponent: b⁻ⁿ = 1 ÷ bⁿ
Fractional exponent: b^(1/n) = the nth root of b
Calculating 2 to the power of 10.
Base: 2
Exponent: 10
Multiply 2 by itself ten times
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Answer: 2¹⁰ = 1,024
A power calculator raises a base to an exponent. It handles whole-number powers such as 2¹⁰, negative powers that give reciprocals, and fractional or decimal powers that give roots or growth factors. You enter the base and the exponent, and it returns the value of baseᵉˣᵖᵒⁿᵉⁿᵗ.
Powers are everywhere once you look. Compound interest multiplies a sum by (1 + r) once per year, so ten years of 8% growth is 1.08¹⁰. Computer memory comes in powers of two. Area uses squares, volume uses cubes, and scientific notation uses powers of ten. The calculator is useful whenever repeated multiplication would be slow by hand, or when the exponent is not a whole number at all.
1. For a positive whole exponent n, multiply the base b by itself n times: bⁿ = b × b × ... × b.
2. Speed up large whole exponents by repeated squaring, such as b⁸ = ((b²)²)².
3. For exponent 0, the answer is 1 for any non-zero base.
4. For a negative exponent, use b⁻ⁿ = 1 ÷ bⁿ.
5. For a fractional exponent p/q, take the q-th root and raise it to the power p: b^(p/q) = (q-th root of b)ᵖ.
6. For any decimal exponent, use logarithms: bˣ = 10^(x × log b), or use a calculator.
Since bᵐ is m copies of b multiplied together, multiplying bᵐ by bⁿ gives m + n copies, so bᵐ × bⁿ = bᵐ⁺ⁿ. Dividing removes copies, so bᵐ ÷ bⁿ = bᵐ⁻ⁿ. Raising to another power repeats the whole block, so (bᵐ)ⁿ = bᵐⁿ. Every other rule is forced by keeping these laws true. For example, b³ ÷ b³ = 1 but also b⁰, so b⁰ = 1. And b⁰ ÷ bⁿ = b⁻ⁿ, so b⁻ⁿ = 1 ÷ bⁿ.
If (b^(1/2))² must equal b¹ by the power law, then b^(1/2) is a number that squares to b, which is √b. In the same way b^(1/3) is the cube root. A decimal exponent such as 1.5 means b^(3/2), the square root cubed. For a negative base, fractional powers are usually not real numbers, so (−8)^(1/3) may show as undefined in software even though the real cube root is −2. Use the cube root calculator for such cases.
A quantity that grows by r per period is multiplied by (1 + r) each time, so after n periods it is multiplied by (1 + r)ⁿ. With a base of 1.08 and exponent 10, the power gives the factor by which money grows over ten years at 8% compounded yearly. Small differences in the base add up: 1.07¹⁰ is about 1.97 while 1.09¹⁰ is about 2.37. Bases between 0 and 1 give decay instead, such as depreciation of a machine or a car.
Lakshmi wants the factor by which a sum grows over 10 years at 8% compounded once a year, so she raises 1.08 to the power 10.
Answer: Result 2.1589249973
Multiplying base by exponent, such as treating 2¹⁰ as 20.
Believing a negative exponent gives a negative answer; 2⁻³ is 0.125, not −8.
Writing −3² when (−3)² is meant; the first is −9 and the second is 9.
Adding exponents when bases differ, such as 2³ × 3² = 6⁵, which is false.
Expecting 0⁰ to have a single agreed value in every context.
Compound interest and investment growth factors.
Computing sizes in binary, such as 2¹⁰ = 1,024 bytes in a kibibyte.
Areas and volumes, which use squares and cubes.
Depreciation and radioactive decay with a base below 1.
Checking scientific notation and orders of magnitude.
Why is anything to the power of 0 equal to 1?
Because dividing powers subtracts the exponents: 2³ ÷ 2³ = 2⁰. Anything divided by itself equals 1, so 2⁰ must also equal 1. The one exception is 0⁰, which is left undefined.
What does a fractional exponent mean?
It means a root. b^(1/2) is the square root, b^(1/3) is the cube root, and b^(2/3) means the cube root, squared.