^ = use e together base

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### What is an exponent?

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

You are watching: 10 to the power of 4

an = a × a × ... × a n times

The jajalger2018.org over accepts an unfavorable bases, yet does no compute imaginary numbers. It also does not accept fractions, but can be offered to compute fountain exponents, as lengthy as the exponents room input in their decimal form.

### Basic exponent laws and also rules

When exponents the share the very 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 one exponent is negative, the an unfavorable sign is removed by reciprocating the base and also raising it to 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 room 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 exponents are raised 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 elevated to one exponent, the exponent is distributed to both bases.

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

Similarly, when split bases are elevated to one exponent, the exponent is distributed 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 remains the same.

a1 = a

When an exponent is 0, the an outcome of the indices of any kind of base will constantly be 1, back somedebate surrounding 00 being 1 or undefined. For plenty of applications, specifying 00 as 1 is convenient.

a0 = 1

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

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

Thus, the only means for an to remain unchanged by multiplication, and also this exponent law to continue to be true, is because that a0 to be 1.

When an exponent is a fraction where the numerator is 1, the nth root of the basic is taken. Shown below is an instance with a spring exponent wherein the numerator is no 1. It offers both the preeminence displayed, as well as the ascendancy for multiplying index number with prefer bases discussed above. Keep in mind that the jajalger2018.org have the right to calculate fractional exponents, yet they need to be gone into into the jajalger2018.org in decimal form.

See more: What Is The Square Root Of 16, How To Find The Square Root Of 16

It is also possible to compute exponents with negative bases. They follow much the very same rules as exponents with optimistic bases. Index number with negative bases raised to hopeful integers space equal come their confident counterparts in magnitude, yet vary based upon sign. If the exponent is an even, confident integer, the values will be equal regardless that a positive or an adverse base. If the exponent is an odd, hopeful integer, the result will again have the very same magnitude, however will be negative. When the rules for fractional index number with an unfavorable bases space the same, they involve the usage of imaginary numbers due to the fact that it is not possible to take any kind of root of a negative number. An example is detailed below for reference, but please keep in mind that the jajalger2018.org detailed cannot compute imaginary numbers, and also any inputs that an outcome in an imagine number will certainly return the result "NAN," signifying "not a number." The numerical systems is basically the same as the situation with a optimistic base, other than that the number must be denoted together imaginary.