^=
use e as base
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What is one exponent?

Exponentiation is a mathematical operation, written as an, including the basic a and also an exponent n. In the situation where n is a hopeful integer, exponentiation corresponds to repeated multiplication the the base, n times.

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an = a × a × ... × a n times

The aramuseum.org above accepts an unfavorable bases, yet does not compute imagine numbers. It additionally does no accept fractions, but can be provided to compute spring exponents, as lengthy as the exponents are input in their decimal form.

Basic exponent laws and also rules

When exponents the share the same base space multiplied, the exponents room added.

an × am = a(n+m)EX:22 × 24 = 4 × 16 = 64 22 × 24 = 2(2 + 4) = 26 = 64

When one exponent is negative, the an unfavorable sign is eliminated by reciprocating the base and raising it to the hopeful exponent.

a(-n)=1
an
EX: 2(-3) = 1 ÷ 2 ÷ 2 ÷ 2 =1
8
EX: 2(-3)=1
23
=1
8

When exponents the share the same base space divided, the exponents are subtracted.

am
an
= a(m - n)
EX: 22
24
=4
16
=1
4
22
24
= 2(2-4) = 2-2 =1
22
=1
4

When index number are elevated to one more exponent, the exponents space multiplied.

(am)n = a(m × n)EX: (22)4 = 44 = 256(22)4 = 2(2 × 4) = 28 = 256

When multiply bases are increased to one exponent, the exponent is dispersed to both bases.

(a × b)n = one × bnEX: (2 × 4)2 = 82 = 64(2 × 4)2 = 22 × 42 = 4 × 16 = 64

Similarly, when divided bases are elevated to an exponent, the exponent is dispersed to both bases.

(a
b
)n=an
bn
EX: (2
5
)2=2
5
×2
5
=4
25
(2
5
)2=22
52
=4
25

When one exponent is 1, the base remains the same.

a1 = a

When one exponent is 0, the result of the exponentiation of any kind of base will always be 1, back somedebate surrounds 00 being 1 or undefined. For plenty of applications, specifying 00 as 1 is convenient.

a0 = 1

Shown listed below is an example of an argument for a0=1 using among the abovementioned exponent laws.

If one × to be = a(n+m)Thenan × a0 = a(n+0) = an

Thus, the only method for an to continue to be unchanged by multiplication, and this exponent regulation to continue to be true, is for a0 to be 1.

When one exponent is a portion where the numerator is 1, the nth root of the basic is taken. Shown below is an example with a fountain exponent wherein the molecule is not 1. It offers both the dominance displayed, and also the dominion for multiplying exponents with like bases debated above. Note that the aramuseum.org can calculate fractional exponents, but they should be entered into the aramuseum.org in decimal form.

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It is also possible to compute exponents with an adverse bases. They follow much the exact same rules together exponents with confident bases. Index number with negative bases raised to hopeful integers space equal come their hopeful counterparts in magnitude, yet vary based upon sign. If the exponent is one even, optimistic integer, the values will be equal regardless that a confident or an adverse base. If the exponent is one odd, hopeful integer, the an outcome will again have the same magnitude, but will it is in negative. When the rules for fractional exponents with negative bases room the same, castle involve the use of imaginary numbers since it is not feasible to take any root the a an adverse number. An example is listed below because that reference, but please note that the aramuseum.org listed cannot compute imagine numbers, and any inputs that an outcome in an imaginary number will return the an outcome \"NAN,\" signifying \"not a number.\" The numerical solution is basically the exact same as the instance with a optimistic base, other than that the number must be denoted as imaginary.