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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.


Exponentiation (Power) -

Tag(s) : Arithmetics

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Exponentiation (Power)

Exponentiation Calculator a^b

Exponentiation Simplifier

Modular Exponentiation Calculator a^b mod n

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

⮞ Go to: Knuth's Arrows

Answers to Questions (FAQ)

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:


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.

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