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

Beta version#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
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Symbolic algebra

ⓘ Hint: This calculator supports symbolic math. You can enter numbers, but also symbols like a, b, pi or even whole math expressions such as (a+b)/2. If you still don't sure how to make your life easier using symbolic algebra check out our another page: Symbolic calculations

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

\left(x+y\right)^{2}

Power in the simplest form

Show source

x^{2}+2~x \cdot y+y^{2}

Exponentiation step-by-step

1	Show source $\left(x+y\right)^{2}$	Short multiplication formula	Short multiplication formula for square of sum: $\left(a+\mathrm{b}\right)^{2}=a^{2}+2~a \cdot \mathrm{b}+\mathrm{b}^{2}$
2	Show source $x^{2}+2~x \cdot y+y^{2}$	Result	Your expression reduced to the simplest form known to us.

Some facts#

Exponentiation means multiplying the same number by itself many times:

$a^n = \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{n}$
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: $2^3 = 2 \cdot 2 \cdot 2 = 8$
We read symbol $2^3$ 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, $2^3$ is a different number than $3^2$ .

⚠ WARNING! $a^n \ne n^a$
Raising any number to the first power does not change the value. For example, $3^1$ is 3:

ⓘ Remember: $a^1 = a$
In turn raising to zero power gives number one e.g. $3^0$ gives 1:

ⓘ Remember: $a^0 = 1$
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. $x^{-1}$ 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: $\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$
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: $a^{\frac{p}{q}} = \left(\sqrt[q]{a}\right)^p = \sqrt[q]{a^p}$
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:

$0^n = \underbrace{0 \cdot 0 \cdot 0 \cdot \ldots \cdot 0}_{n} = 0$

$1^n = \underbrace{1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1}_{n} = 1$
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:

$\left(-a\right)^n = \left\{ \begin{array}{ll} a^n & \textrm{if n is even}\\ -a^n & \textrm{if n is odd}\\ \end{array} \right.$

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, $\TeX$ , $\LaTeX$ , 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.

Tags and links to this website#

Tags:

power_calculator · power_simplify · exponentiation_calculator · x_pow_y

Tags to Polish version:

potega · potegowanie_kalkulator · x_pow_y · najprostsza_postac_potegi · doprowadz_potege_do_najprostszej_postaci · kalkulator_potegowania

What tags this calculator has#

CALCULATOR · MATH · A -> Z (all)

Permalink#

This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

https://calculla.com/power_calculator?inBase=$symbolic(x @2B; y)&inExponent=$symbolic(2)

Links to external sites (leaving Calculla?)#