^=
use e as base
*

RelatedScientific altoalsimce.org | log altoalsimce.org | root altoalsimce.org

What is an exponent?

Exponentiation is a mathematics operation, composed as an, entailing the base a and also an exponent n. In the situation where n is a optimistic integer, exponentiation synchronizes to repeated multiplication that the base, n times.

You are watching: 2 to the power of -5

an = a × a × ... × a n times

The altoalsimce.org above accepts an unfavorable bases, but does not compute imaginary numbers. It also does no accept fractions, however can be supplied to compute fractional exponents, as lengthy as the exponents space input in their decimal form.

Basic exponent laws and rules

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

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

When an exponent is negative, the negative sign is eliminated by reciprocating the base and also raising it come the optimistic 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 room 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 raised to an additional exponent, the exponents room multiplied.

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

When multiplied bases are raised to an exponent, the exponent is distributed 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 an exponent is 1, the base continues to be the same.

a1 = a

When one exponent is 0, the an outcome of the indexes of any type of base will always be 1, although somedebate surrounds 00 gift 1 or undefined. For plenty of applications, defining 00 as 1 is convenient.

a0 = 1

Shown below is an instance of an debate for a0=1 using among the abovementioned exponent laws.

If an × am = a(n+m)Thenan × a0 = a(n+0) = an

Thus, the only means for an to continue to be unchanged by multiplication, and also this exponent regulation to stay true, is for a0 to it is in 1.

When an exponent is a portion where the molecule is 1, the nth root of the base is taken. Shown listed below is an example with a fractional exponent whereby the molecule is no 1. It provides both the preeminence displayed, and the preeminence for multiplying exponents with favor bases questioned above. Note that the altoalsimce.org have the right to calculate spring exponents, however they should be gotten in into the altoalsimce.org in decimal form.

See more: Does King Consider Himself An Outsider, Kings Speech From Birmingham Jail

*

It is also feasible to compute index number with negative bases. They follow lot the very same rules together exponents with positive bases. Exponents with negative bases elevated to hopeful integers are equal to their optimistic counterparts in magnitude, yet vary based upon sign. If the exponent is one even, confident integer, the worths will be equal regardless of a optimistic or negative base. If the exponent is one odd, positive integer, the result will again have actually the very same magnitude, but will be negative. When the rules because that fractional exponents with negative bases space the same, castle involve the use of imaginary numbers because it is not possible to take any root that a an unfavorable number. An instance is listed below because that reference, however please keep in mind that the altoalsimce.org provided cannot compute imagine numbers, and any inputs that an outcome in an imagine number will certainly return the an outcome "NAN," signifying "not a number." The numerical solution is essentially the very same as the situation with a positive base, except that the number must be denoted together imaginary.