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 - dCode

Tag(s) : Arithmetics

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

Go to: Modular Exponentiation

Go to: Knuth's Arrows

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.

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

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

__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 negative power 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 formal calculator on dCode.

The exponentiation operator has 3 main mathematical identities:

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

dCode retains ownership of the source code of the script Exponentiation online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Exponentiation script for offline use on PC, iPhone or Android, ask for price quote on contact page !

- Exponentiation Calculator a^b
- Modular Exponentiation Calculator a^b mod n
- 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 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?

exponentiation,power,exponent,multiplication,calculus,base,exponential,circumflex,calculator,arithmetic

Source : https://www.dcode.fr/exponentiation-calculation

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲