The simplest power form calculator
Calculator finds out the simplest form of given power (exponentiation) expression. Shows all the step-by-step progress of simplification.

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Input data - power base and exponent

Power base
(expression, which we're going to raise to power)
Exponent
(the degree of the power, we're going to raise the base to this number)

Results - your expression in the simplest form

Power, which you enter
Show source(x+y)2{\left( x+ y\right)}^{2}
Power in the simplest form
Show sourcex2+2 xy+y2{ x}^{2}+2~ x\cdot y+{ y}^{2}
Exponentiation step-by-step
1Show source(x+y)2{\left( x+ y\right)}^{2}Short multiplication formula for square of sum:
(a+b)2=a2+2 a b+b2{\left( a+\mathrm{b}\right)}^{2}={ a}^{2}+2~ a~\mathrm{b}+{\mathrm{b}}^{2}
2Show sourcex2+2 xy+y2{ x}^{2}+2~ x\cdot y+{ y}^{2}Result

Some facts

  • Exponentiation means multiplying the same number by itself many times:
    an=aaaana^n = \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{n}
    where:
    • a - the base of exponentiation, it's a number which we multiply by itself,
    • n - the exponent of exponentiation, it's a number of multiplications performed.
    ⓘ Example: 23=222=82^3 = 2 \cdot 2 \cdot 2 = 8
  • We read symbol 232^3 as "two to the third power" or more colloquially: "two to third".
  • Formally, the exponentiation is a two-argument operation, where the first argument is the base (number 2 in above example) and the second one is exponent (number 3 in the above example).
  • Exponentiation is not commutable, i.e. you can not swap the base with the exponent. For example, 232^3 is a different number than 323^2.
    ⚠ WARNING! annaa^n \ne n^a
  • Raising any number to the first power does not change the value. For example, 313^1 is 3:
    ⓘ Remember: a1=aa^1 = a
  • In turn raising to zero power gives number one e.g. 303^0 gives 1:
    ⓘ Remember: a0=1a^0 = 1
  • Exponentiation by the negative number is the same as performing an identical operation, but with inverse base. Therefore, often the inverse operation is written as an increase to the power of -1, e.g. x1x^{-1} means as much as "the inverse of x". If you want to know more about inverse of the numbers, then you can check out our other calculator: Fractions: inverse (reciprocal). In general, the folowing formula is met
    ⓘ Remember: (ab)n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
  • Exponentiation with a non-integer exponent is the same as root. The exponentiation with rational exponent may be used to present root and power in one operation. In general, the following formula is met:
    ⓘ Remember: apq=(aq)p=apqa^{\frac{p}{q}} = \left(\sqrt[q]{a}\right)^p = \sqrt[q]{a^p}

  • Basing on above formula we can see that the power and root are in fact the same kind of operation. In practice, this means that it does not matter in what order we perform these operation (we can calculate root first and raise to the power next or vice versa). In both cases we will get the same result.
  • Raising the number one to any power gives one. Similarly, zero to any power gives zero. These facts result from property of multiplication by one and zero:
    0n=0000n=00^n = \underbrace{0 \cdot 0 \cdot 0 \cdot \ldots \cdot 0}_{n} = 0
    1n=1111n=11^n = \underbrace{1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1}_{n} = 1
  • If you want to learn more about elementary math operations such as multiplication you can check out our other calculator: Number operations.
  • If the exponentiation base is negative, then the sign of the result depends on parity of exponent. Even exponents give positive result and odd exponents give the negative one. In general, we can write:
    (a)n={anif n is evenanif n is odd \left(-a\right)^n = \left\{ \begin{array}{ll} a^n & \textrm{if n is even}\\ -a^n & \textrm{if n is odd}\\ \end{array} \right.

Computers and programming languages

  • In some programming languages​​ exponentiation is built into the syntax of the language itself, e.g.:
    • Algol, Commodore BASIC:
      x ↑ y

    • AWK, BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX\TeX, LaTeX\LaTeX, TI-BASIC, bc, Haskell, Lua and Calculla:
      x ^ y

    • Haskell (for integer exponents), D:
      x ^^ y

    • Ada, Z shell, Korn shell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, Groovy, JavaScript, OCaml, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for float exponents), Turing, VHDL:
      x ** y
  • In programmed languages ​​in which exponentiation is not an integral part of the syntax, the exponential is accomplished by call to the library function e.g.
    • C, C++:
      pow(x, y)

    • C#:
      Math.Pow(x, y)

    • Erlang, Javascript:
      Math.pow(x, y)
  • Examples showing how exponentiation can be implemented on-by-own (without library functions or built-in operator if exists) in various programming languages can be found on rosettacode.org site.

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