Tool to compute power of a number. Exponentiation (or power) of a number 'a^b' is the result of the 'b'-times repeated multiplication of the number 'a' by itself.

Exponentiation (Power) - dCode

Tag(s) : Arithmetics

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⮞ Go to: Modular Exponentiation

⮞ Go to: Knuth's Arrows

Calculating $ a $ power $ b $ (also called $ a $ exponent $ b $ or $ a $ exponential $ b $) corresponds to multiply $ a $ by itself $ b $ times.

$$ a^n = \underbrace{a \times \cdots \times a}_{b} $$

__Example:__ $$ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 $$

In a power calculation $ a^b = c $, the part $ a $ is called the *base* and the part $ b $ is called the *exponent*, $ c $ is normally called *power* but in everyday language *power* and *exponentiation* are often exchanged.

Like the multiplication tables, there is a, exponentiation table, or table of powers but this one is not symmetrical (a^b is not always equal to b^a). Here is the table reading `row^column`:

\ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 |

3 | 1 | 3 | 9 | 27 | 81 | 243 | 729 | 2187 | 6561 | 19683 | 59049 |

4 | 1 | 4 | 16 | 64 | 256 | 1024 | 4096 | 16384 | 65536 | 262144 | 1048576 |

5 | 1 | 5 | 25 | 125 | 625 | 3125 | 15625 | 78125 | 390625 | 1953125 | 9765625 |

6 | 1 | 6 | 36 | 216 | 1296 | 7776 | 46656 | 279936 | 1679616 | 10077696 | 60466176 |

7 | 1 | 7 | 49 | 343 | 2401 | 16807 | 117649 | 823543 | 5764801 | 40353607 | 282475249 |

8 | 1 | 8 | 64 | 512 | 4096 | 32768 | 262144 | 2097152 | 16777216 | 134217728 | 1073741824 |

9 | 1 | 9 | 81 | 729 | 6561 | 59049 | 531441 | 4782969 | 43046721 | 387420489 | 3486784401 |

10 | 1 | 10 | 100 | 1000 | 10000 | 100000 | 1000000 | 10000000 | 100000000 | 1000000000 | 10000000000 |

In mathematics, a power is written $ a $ `exponent` $ b $ as $ a^b $.

In computing, the exponentiation calculation is often indicated by a circumflex `^` : `a^b` for `a` power `b`. In some programming languages, the notation `a**b` (the multiplication sign twice) is used.

A power of 0 is equal to 1. Any number exponent 0 is worth 1. $$ a^0 = 1 $$

0 to the power of 0 is a debatable case but by convention, $ 0^0 = 1 $

A power with a negative exposant is the inverse of a positive power.

$$ a^{-n} = \frac{1}{a^n} $$

__Example:__ $ 2^{-3} = \frac{1}{2^3} $

The power $ -1 $ in maths is equivalent to a mathematical inverse.

$$ n^{-1} = \frac{1}{n^1} = \frac{1}{n} $$

__Example:__ $ 2^{-1} = \frac{1}{2} $

To calculate the $ x $ last digits of a power $ a^b $, use the modular exponentiation calculator $ a^b \mod 10^x $

__Example:__ Finding the last 3 digits of $ 2^20 = 1048576 $ is to calculate $ 2^20 \mod 10^3 = 576 $ (NB: $ 10^3 = 1000 $)

Exponent corresponds to the number of times the multiplication has to be done. If the exponent is not an integer, it becomes a root calculus and is not managed by this function. Use the square root page or the formal calculator on dCode.

The exponentiation operator has 3 main mathematical identities:

$$ a^{b + c} = a^b \cdot a^c \\ (a^b)^n = a^{b \cdot n} \\ (a \cdot b)^n = a^n \cdot b^n $$

Note also that an even power of a negative number is always positive, and an odd power of a negative number is always negative.

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*Exponentiation (Power)* on dCode.fr [online website], retrieved on 2022-11-28,

- Exponentiation Calculator a^b
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- Iterated Exponentiation Calculator a^a^...^a
- How to calculate a raised to power b?
- What are the base and the exponent in an exponentiation? (Definition)
- What are power tables?
- How to write a raised to power b?
- What is a^0 (power zero)?
- What is a negative power?
- What is the minus one -1 power?
- How to calculate the last digits of an exponentiation?
- Why exponent has to be an integer and not rational?
- What are exponentiation operation properties?

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