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What is an exponent?
Exponentiation is a mathematical operation, composed as an, involving the base a and an exponent n. In the case where n is a hopeful integer, exponentiation synchronizes to repeated multiplication of the base, n times.
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an = a × a × ... × a n times
The rebab.net above accepts negative bases, but does no compute imagine numbers. It additionally does not accept fractions, but can be supplied to compute fractional exponents, as lengthy as the exponents space input in your decimal form.
Basic exponent laws and also rules
When exponents the share the exact same base room 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 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 |
8 |
When exponents the share the same base room divided, the exponents are subtracted.
am |
an |
EX: | 22 |
24 |
16 |
4 |
22 |
24 |
22 |
4 |
When index number 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 increased to an exponent, the exponent is spread to both bases.
(a × b)n = one × bnEX: (2 × 4)2 = 82 = 64(2 × 4)2 = 22 × 42 = 4 × 16 = 64
Similarly, when split bases are increased to an exponent, the exponent is spread to both bases.
( | a |
b |
bn |
EX: ( | 2 |
5 |
5 |
5 |
25 |
( | 2 |
5 |
52 |
25 |
When an exponent is 1, the base stays the same.
a1 = a
When one exponent is 0, the result of the indices of any base will always be 1, back somedebate surrounding 00 gift 1 or undefined. For numerous applications, defining 00 together 1 is convenient.
a0 = 1
Shown below is an instance of an debate for a0=1 using one of the aforementioned exponent laws.
If one × am = a(n+m)Thenan × a0 = a(n+0) = an
Thus, the only way for an to continue to be unchanged through multiplication, and this exponent legislation to stay true, is because that a0 to be 1.
When one exponent is a fraction where the numerator is 1, the nth source of the base is taken. Shown below is an instance with a fountain exponent wherein the molecule is not 1. It supplies both the rule displayed, and the dominion for multiplying index number with choose bases debated above. Keep in mind that the rebab.net have the right to calculate fractional exponents, however they have to be gone into into the rebab.net in decimal form.
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It is also feasible to compute index number with negative bases. They follow much the exact same rules as exponents with confident bases. Exponents with an unfavorable bases raised to hopeful integers space equal to their hopeful counterparts in magnitude, yet vary based on sign. If the exponent is one even, hopeful integer, the worths will be equal regardless the a optimistic or an adverse base. If the exponent is an odd, confident integer, the result will again have actually the exact same magnitude, yet will it is in negative. While the rules for fractional index number with negative bases space the same, castle involve the use of imaginary numbers due to the fact that it is not possible to take any type of root that a negative number. An example is detailed below for reference, yet please keep in mind that the rebab.net detailed cannot compute imagine numbers, and also any input that result in an imaginary number will return the an outcome "NAN," signifying "not a number." The numerical equipment is essentially the same as the situation with a optimistic base, other than that the number need to be denoted together imaginary.