Powers: repeated multiplication
A power is shorthand for repeated multiplication. 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The lower number is called the base, the raised number the exponent or index. In the online calculator you enter powers with the x^y key or the ^ character on your keyboard: 2^5 gives 32.
Square and cube
The most common powers have their own keys. x² forms the square, x³ the cube. You need square numbers for areas: a square room with a side length of 4.5 metres has 4.5² = 20.25 square metres. The cube is found in volumes: a cube with an edge length of 30 centimetres holds 30³ = 27,000 cubic centimetres, i.e. 27 litres. For more complex shapes, the area calculator and the volume calculator help.
Negative and fractional exponents
A negative exponent means reciprocal: 2^(−3) = 1 ÷ 2³ = 0.125. The 1/x key does nothing other than append the power −1. A fractional exponent stands for a root: 9^(1/2) is the square root of 9, i.e. 3, and 8^(1/3) is the cube root of 8, i.e. 2. Any power with the exponent 0 equals 1, for example 7^0 = 1.
Taking roots
The square root is the inverse of the square: √81 = 9, because 9² = 81. Press √ and then the number; the calculator closes the bracket automatically if needed. For the cube root there is the ∛ key: ∛27 = 3. Any nth root can be calculated via the fractional exponent. The fifth root of 32 is 32^(1/5) = 2.
A practical example is Pythagoras' theorem. A television 1.20 metres wide and 0.68 metres high has a diagonal of √(1.2² + 0.68²) = 1.38 metres. Distances between two points or the length of a staircase are calculated in the same way.
Roots of negative numbers
The square root of a negative number is not defined in the real numbers, because no square is negative. √(−4) therefore leads to an error message. Odd roots are different: ∛(−8) = −2, because (−2)³ = −8. The calculator also takes this into account with the notation (−8)^(1/3).
Rules for powers
Powers are calculated before multiplication, division, addition and subtraction. 3 × 2² is 3 × 4 = 12, not 6² = 36. Several powers in a row are calculated from right to left: 2^3^2 = 2^9 = 512. And the minus sign in front of a number binds less strongly than the power: −3² = −9, but (−3)² = 9.
The laws of exponents are also useful. With the same base, exponents are added when multiplying: 2³ × 2⁴ = 2⁷ = 128. When dividing, they are subtracted: 10⁶ ÷ 10² = 10⁴. A power of a power gives the product of the exponents: (5²)³ = 5⁶ = 15,625.
Large numbers and scientific notation
Powers grow quickly. 2^64 is already a number with 20 digits. The calculator therefore shows very large results in scientific notation: 1.84467440737 × 10^19. You can continue calculating directly with this representation. If a result exceeds the representable range, the note "Number too large" appears.
Powers in everyday life: growth and compound interest
Wherever something grows by a percentage, powers are involved. 10,000 at 4% interest over 15 years becomes 10,000 × 1.04^15 = 18,009.44. Inflation works the other way round: at 2% inflation per year, one unit of currency has only 1 ÷ 1.02^20 = 0.67 of its purchasing power after 20 years. Detailed tables are produced by the compound interest calculator and the inflation calculator.