Exponentiation (Power)

Question 1

How to calculate a raised to power b?

Answer

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

Question 2

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

Answer

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.

Question 3

What are power tables?

Answer

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

Question 4

How to write a raised to power b?

Answer

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.

Question 5

What is a^0 (power zero)?

Answer

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 $

Question 6

What is a negative power?

Answer

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} $

Question 7

What is the minus one -1 power?

Answer

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} $

Question 8

How to calculate the last digits of an exponentiation?

Answer

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 $)

Question 9

Why exponent has to be an integer and not rational?

Answer

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.

Question 10

What are exponentiation operation properties?

Answer

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.

Exponentiation (Power)

Exponentiation Calculator a^b

Exponentiation Simplifier

Modular Exponentiation Calculator a^b mod n

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

Answers to Questions (FAQ)