^ | = | ||
use e as base | |||
RelatedScientific slrfc.org | Log slrfc.org | Root slrfc.org
What is an exponent?
Exponentiation is a mathematical operation, created as an, involving the base a and an exponent n. In the instance wbelow n is a positive integer, exponentiation synchronizes to repetitive multiplication of the base, n times.
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an = a × a × ... × a n times
The slrfc.org over accepts negative bases, however does not compute imaginary numbers. It also does not accept fractions, but can be provided to compute fractional exponents, as lengthy as the exponents are input in their decimal form.
Basic exponent laws and also rules
When exponents that share the exact same base are multiplied, the exponents are added.
an × am = a(n+m)EX:22 × 24 = 4 × 16 = 64 22 × 24 = 2(2 + 4) = 26 = 64
When an exponent is negative, the negative authorize is removed by reciprocating the base and also elevating it to the positive exponent.
a(-n)= | 1 |
an |
EX: 2(-3) = 1 ÷ 2 ÷ 2 ÷ 2 | = | 1 |
8 |
EX: 2(-3)= | 1 |
23 |
8 |
When exponents that share the very same base are split, the exponents are subtracted.
am |
an |
EX: | 22 |
24 |
16 |
4 |
22 |
24 |
22 |
4 |
When exponents are raised to an additional exponent, the exponents are 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 dispersed to both bases.
(a × b)n = an × bnEX: (2 × 4)2 = 82 = 64(2 × 4)2 = 22 × 42 = 4 × 16 = 64
Similarly, as soon as split bases are elevated to an exponent, the exponent is dispersed to both bases.
( | a |
b |
bn |
EX: ( | 2 |
5 |
5 |
5 |
25 |
( | 2 |
5 |
52 |
25 |
When an exponent is 1, the base continues to be the very same.
a1 = a
When an exponent is 0, the result of the exponentiation of any kind of base will always be 1, although somedebate surrounds 00 being 1 or undefined. For many applications, specifying 00 as 1 is convenient.
a0 = 1
Shown below is an example of an argument for a0=1 using among the previously mentioned exponent regulations.
If an × am = a(n+m)Thenan × a0 = a(n+0) = an
Hence, the only way for an to reprimary unadjusted by multiplication, and this exponent legislation to reprimary true, is for a0 to be 1.
When an exponent is a portion wbelow the numerator is 1, the nth root of the base is taken. Shown below is an example through a fractional exponent wbelow the numerator is not 1. It supplies both the dominion displayed, as well as the preeminence for multiplying exponents via like bases disputed above. Note that the slrfc.org have the right to calculate fractional exponents, however they have to be entered into the slrfc.org in decimal develop.
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It is also feasible to compute exponents with negative bases. They follow a lot the very same rules as exponents via positive bases. Exponents through negative bases raised to positive integers are equal to their positive counterparts in magnitude, however differ based on authorize. If the exponent is an also, positive integer, the worths will be equal regardless of a positive or negative base. If the exponent is an odd, positive integer, the outcome will certainly aget have actually the exact same magnitude, yet will be negative. While the rules for fractional exponents with negative bases are the exact same, they involve the use of imaginary numbers given that it is not possible to take any type of root of an adverse number. An example is provided below for reference, but please note that the slrfc.org gave cannot compute imaginary numbers, and also any inputs that bring about an imaginary number will rerotate the result "NAN," signifying "not a number." The numerical solution is fundamentally the very same as the situation via a positive base, except that the number have to be delisted as imaginary.