Search for a tool
Exponentiation (Power)

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.

Results

Exponentiation (Power) -

Tag(s) : Arithmetics

Share dCode and more

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!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Exponentiation (Power)' tool, so feel free to write! Thank you!

# Exponentiation (Power)

## Iterated Exponentiation Calculator a^a^...^a

⮞ Go to: Knuth's Arrows

### How to calculate a raised to power b?

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

### What are the base and the exponent in an exponentiation? (Definition)

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.

### What are power tables?

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 \ 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 4 8 16 32 64 128 256 512 1024 1 3 9 27 81 243 729 2187 6561 19683 59049 1 4 16 64 256 1024 4096 16384 65536 262144 1048576 1 5 25 125 625 3125 15625 78125 390625 1953125 9765625 1 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 1 7 49 343 2401 16807 117649 823543 5764801 40353607 282475249 1 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 1 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000

### How to write a raised to power b?

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.

### What is a^0 (power zero)?

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, $0^0 = 1$

### What is a negative power?

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

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

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

### What is the minus one -1 power?

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

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

Example: $2^{-1} = \frac{1}{2}$

### How to calculate the last digits of an exponentiation?

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: $10^3 = 1000$)

### Why exponent has to be an integer and not rational?

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.

### What are exponentiation operation properties?

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.

## Source code

dCode retains ownership of the online "Exponentiation (Power)" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Exponentiation (Power)" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Exponentiation (Power)" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, copy-paste, or API access for "Exponentiation (Power)" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!