Exponential and Exponential

Exponential

Exponential a mathematical binary operation performed on two numbers a and b, the result of the exponentiation operation is the product of the multiplication with b factor a multiply each other. Exponential power is denoted by a^b, read as b power of a, the number a is called the base, the number b is called the exponent.
The operation that is the opposite of exponentiation is the root operation. Exponential (from Sino-Vietnamese: 累乘) means "to multiply". Especially
a² also called “a squared”;
a³ Also called "a cube".

Powers with integer exponents

Powers of zero and one

{\displaystyle 0^{n}=0\,}.
{\displaystyle 1^{n}=1\,}.

Powers with positive integer exponents

In case b = n is a positive integer, the nth power of a is the product of n equal factors, each of which is equal to a:
{\displaystyle a^{n}=\underbrace {a\times a\cdots \times a} _{n}}
Most important properties of powers with positive integer exponents m, n to be
{\displaystyle a^{m+n}=a^{m}\times a^{n}}
{\displaystyle a^{m-n}={\frac {a^{m}}{a^{n}}}} with everyone a ≠ 0
{\displaystyle (a^{m})^{n}=a^{mn}}
{\displaystyle a^{m^{n}}=a^{(m^{n})}}
{\displaystyle (a\times b)^{n}=a^{n}\times b^{n}}
{\displaystyle ({\frac {a}{b}})^{n}={\frac {a^{n}}{b^{n}}}}
In particular, we have:
{\displaystyle a^{1}=a}
While addition and multiplication are commutative, exponentiation is not commutative. Similarly, addition and multiplication are associative, but exponentiation is not.. Without parentheses, the order of powers is from top to bottom, not bottom to top:
{\displaystyle a^{b^{c}}=a^{(b^{c})}\neq (a^{b})^{c}=a^{(b\cdot c)}=a^{b\cdot c}}

Powers with exponent 0

Powers to the zero exponent of the number a other is not conventionally equal to 1.
{\displaystyle a^{0}=1}
Prove
{\displaystyle 1={\frac {a^{n}}{a^{n}}}=a^{n-n}=a^{0}}

Powers with negative exponents

Powers of a with a negative integer exponent m, where ({\displaystyle m=-n}) a non-zero and n is a positive integer:
{\displaystyle a^{-n}={\frac {1}{a^{n}}}}.
For example
{\displaystyle 3^{-4}={\frac {1}{3^{4}}}={\frac {1}{3.3.3.3}}={\frac {1}{81}}}.
How to deduce “power with negative exponent” from “power with zero exponent”:
{\displaystyle a^{0}=a^{n-n}={\frac {a^{n}}{a^{n}}}=a^{n}.{\frac {1}{a^{n}}}=a^{n}.a^{-n}}
Special case, power of non-zero a with the exponent −1 being its reciprocal. {\displaystyle a^{-1}={\frac {1}{a}}.}

Powers of positive real numbers with real exponents

The nth root of a positive real number

One square root n of number a is a number x so that xn = a. If a is a positive real number, n is a positive integer, x If it is not negative, then there is exactly one positive real number x such that xn = a. This number x is called the nth-order arithmetic root of . a.It is denoted by n√a, where √ is the root symbol.

Powers to rational exponents of positive real numbers

Powers with minimalist rational exponents m/n (m, n is an integer, where n is positive), of a positive real number a is defined as
{\displaystyle a^{m/n}=\left(a^{m}\right)^{1/n}={\sqrt[{n}]{a^{m}}}}
This definition can be extended to negative real numbers whenever the root is significant.