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Whole number and integer exponent

Definition and properties

 

Expression of an exponentiation

The exponentiation of whole numbers is the operation that involves the base with repeated multiplication by the same base. Given that ${\displaystyle b}$  is the base, and ${\displaystyle n}$  is the exponent located at the base's superscript, representing how many bases are being multiplied. The result of this operation gives the power:^[1] $${\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.}$$  An example is ${\displaystyle 2^{3}}$ , where one may write the expression as ${\displaystyle 2\times 2\times 2}$ , so the power is 8. When setting ${\displaystyle n=1}$ , the operation results in the power of the base itself, which is ${\displaystyle b^{1}=b}$ .^[2] The base and the exponent is not interchangeable, because the power will be different, like ${\displaystyle 3^{2}=3\times 3=9}$ , thereby the exponentiation is not commutative.^[3]

Addition and subtraction are capable of applying the exponentiation within, forasmuch as the bases are the same. The summation of two powers of the exponentiation results in the product of two bases with different powers. This is known as "multiplication rule", and by the definition above, for different exponents ${\displaystyle m}$  and ${\displaystyle n}$ , one may consider the product of two exponentiation ${\displaystyle b^{m}}$  and ${\displaystyle b^{n}}$  as the multiplication of the ${\displaystyle b}$  with ${\displaystyle m+n}$  factors:^[4] $${\displaystyle {\begin{aligned}b^{m}\cdot b^{n}&=\underbrace {b\times b\times \dots \times b\times b} _{m{\text{ times}}}\times \underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}\\&=\underbrace {b\times b\times \dots \times b\times b} _{m+n{\text{ times}}}\\&=b^{m+n}.\end{aligned}}}$$  Conversely, the difference of exponents can be considered as the division of exponentiation by the cancellation rule:^[4] $${\displaystyle ={\begin{aligned}{\frac {b^{m}}{b^{n}}}&={\frac {\underbrace {b\times b\times \dots \times b\times b} _{m{\text{ times}}}}{\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}}}\\&=\underbrace {b\times b\times \dots \times b\times b} _{m-n{\text{ times}}}\\&=b^{m-n}.\end{aligned}}}$$  For example, ${\displaystyle 2^{3}\times 2^{4}=2^{3+4}=2^{7}}$  and ${\displaystyle 2^{3}\div 2^{1}=2^{3-1}=2^{2}}$ . As for the second, when the power is reduced to zero, the exponentiation gives 1. In other words, ${\displaystyle b^{0}=1}$ .^[5] However, the base must not be zero, or give undefined expressions otherwise, and any base with a zeroth exponent is always zero.^[6]

Other than two basic operations in arithmetic, multiplication includes the product of two different exponents, which is the identity of exponent of the exponentiation:^[4] $${\displaystyle (b^{m})^{n}=b^{m\cdot n}.}$$  An example is ${\displaystyle (2^{3})^{2}=2^{3\times 2}=2^{6}}$ . Nevertheless, this cannot be confused for the identity of exponent is the exponentiation itself ${\displaystyle b^{m^{n}}=b^{(m^{n})}}$ , thereby the exponentiation is not associative: ${\displaystyle (b^{m})^{n}\neq b^{(m^{n})}}$ .^[7] To prevent, the parentheses are conventionally applied for the order of operations; in the first identity, the parenthesis includes the powered base ${\displaystyle b^{m}}$ , whereas the latter is not necessarily to be included.^[8]

The number exponent can be stretched to the set of integers wherein a negative number exists after the set of whole numbers. When the exponent is a negative number, the results form the invertible expression:^[9] $${\displaystyle b^{-n}={\frac {1}{b^{n}}}.}$$  A special case of this property is when setting ${\displaystyle n=1}$ . This applies to symbolize the inversion of a mathematical object. For example, invertible elements of ${\displaystyle x}$  is standardly denoted as ${\displaystyle x^{-1}}$  in algebraic structure,^[10] or invertible function of ${\displaystyle f}$  denotes ${\displaystyle f^{-1}}$  in the field of mathematical analysis.^[11]

Given two different bases ${\displaystyle a}$  and ${\displaystyle b}$ . Considering that the power of those bases is the same, their product is the power of the product of two bases:^[4] $${\displaystyle a^{n}\cdot b^{n}=(a\cdot b)^{n}.}$$ 

Particular bases

Main articles: Power of two, Power of ten, and Binomial formula

A previous example of the particular base in exponentiation is when the base ${\displaystyle b=0}$ , thereby any power of zero ${\displaystyle 0^{n}}$  results in zero, forasmuch as the exponent cannot be zero. Other examples are the base ${\displaystyle b=1}$  in which any power of this base ${\displaystyle 1^{n}}$  is always 1, but the case for the exponent ${\displaystyle n}$  tends to infinity is undefined. For ${\displaystyle b=-1}$ , the results of ${\displaystyle (-1)^{n}}$  are 1 in the case of even exponent and ${\displaystyle -1}$  in the case of odd. For the base ${\displaystyle b=2}$ , ${\displaystyle 2^{n}}$  results from multiplying 2 for the given factors, and the ascertain is commonly applied in set theory as the power set and computer science give the number of possible values for an n-bit integer in binary number system. Power of ten ${\displaystyle 10^{n}}$  is applied in scientific notation as a shorthand for expressing inordinately large or small numbers to be conveniently written in decimal form.

The powers of a sum can normally be computed from the powers of the summands by the binomial formula $${\displaystyle (a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.}$$  However, this formula is true only if the summands commute ${\displaystyle ab=ba}$ , which is implied if they belong to a structure that is commutative. Otherwise, if ${\displaystyle a}$  and ${\displaystyle b}$  are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general-purpose computer algebra systems use a different notation, sometimes ^^ instead of ^ for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

Power function

 

Power functions for ${\displaystyle n=1,3,5}$ 

 

Power functions for ${\displaystyle n=2,4,6}$ 

Real functions of the form ${\displaystyle f(x)=cx^{n}}$ , where ${\displaystyle c\neq 0}$ , are sometimes called power functions.^[12] When ${\displaystyle n}$  is an integer and ${\displaystyle n\geq 1}$ , two primary families exist: for ${\displaystyle n}$  even, and for ${\displaystyle n}$  odd. In general for ${\displaystyle c>0}$ , when ${\displaystyle n}$  is even ${\displaystyle f(x)=cx^{n}}$  will tend towards positive infinity with increasing ${\displaystyle x}$ , and also towards positive infinity with decreasing ${\displaystyle x}$ . All graphs from the family of even power functions have the general shape of ${\displaystyle y=cx^{2}}$ , flattening more in the middle as ${\displaystyle n}$  increases.^[13] Functions with this kind of symmetry ${\displaystyle f(-x)=f(x)}$  are called even functions.^[14]

When ${\displaystyle n}$  is odd, ${\displaystyle f(x)}$ 's asymptotic behavior reverses from positive ${\displaystyle x}$  to negative ${\displaystyle x}$ . For ${\displaystyle c>0}$ , ${\displaystyle f(x)=cx^{n}}$  will also tend towards positive infinity with increasing ${\displaystyle x}$ , but towards negative infinity with decreasing ${\displaystyle x}$ . All graphs from the family of odd power functions have the general shape of ${\displaystyle y=cx^{3}}$ , flattening more in the middle as ${\displaystyle n}$  increases and losing all flatness there in the straight line for ${\displaystyle n=1}$ . Functions with this kind of symmetry ${\displaystyle f(-x)=-f(x)}$  are called odd functions.^[14]

For ${\displaystyle c<0}$ , the opposite asymptotic behavior is true in each case.^[13]

[1]

Rational number exponent

 

From top to bottom: ${\displaystyle x^{1/8}}$ , ${\displaystyle x^{1/4}}$ , ${\displaystyle x^{1/2}}$ , ${\displaystyle x^{1}}$ , ${\displaystyle x^{2}}$ , ${\displaystyle x^{4}}$ , ${\displaystyle x^{8}}$ . The ___domain of all functions is unit interval.

The definition of exponentiation as repeated multiplication is no longer applicable if the exponent is a rational number. This exponent is considered as the fraction, which can be alternatively written as root. Depending on the numerator and denominator of a root, the exponent ${\displaystyle b^{1/n}}$  can be written as ${\displaystyle {\sqrt[{n}]{b}}}$ , and $${\displaystyle b^{m/n}=(b^{m})^{\frac {1}{n}}={\sqrt[{n}]{b^{m}}},}$$  for ${\displaystyle n}$  is natural number.^[3]

[2]

An algebraic number is a number obtained from the root of a polynomial. Therefore, if the base ${\displaystyle b}$  is a positive real algebraic number, and ${\displaystyle n}$  is a rational number, then ${\displaystyle b^{n}}$  is also an algebraic number, which results from the theory of algebraic extensions. This remains true if ${\displaystyle b}$  is any algebraic number, in which case, all values of ${\displaystyle b^{n}}$  (as a multivalued function) are algebraic. If ${\displaystyle n}$  is irrational number, and both ${\displaystyle b}$  and ${\displaystyle n}$  are algebraic, Gelfond–Schneider theorem asserts that all values of ${\displaystyle b^{n}}$  are transcendental (that is, not algebraic), except if ${\displaystyle b}$  equals 0 or 1. In other words, if ${\displaystyle n}$  is irrational and ${\displaystyle b\notin \{0,1\}}$ , then at least one of ${\displaystyle b}$ , ${\displaystyle n}$ , and the exponentiation ${\displaystyle b^{n}}$  is transcendental.

Real number exponent

Footnotes

1. • Musser, Peterson & Burger (2013), p. 117
   • Young (2022), p. 16
2. Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach. John Wiley & Sons. p. 120. ISBN 978-1-118-48700-6.
3. Klose, Orval (1966). The Number Systems and Operations of Arithmetic: An Explanation of the. Pergamon Press. p. 107.
4. Musser, Peterson & Burger (2013), p. 117.
5. Young, Cynthia Y. (2022). Algebra and Trigonometry (5th ed.). John Wiley & Sons. p. 18.
6. Musser, Peterson & Burger (2013), p. 120.
7. Klose (1966), p. 108.
8. Posamentier, Elaine; Farber, William; Germain-Williams, Terri L.; Paris; Thaller, Bernd; Lehmann, Ingmar (2013). 100 Commonly Asked Questions in Math Class. Corwin Press. pp. 71–73. ISBN 978-1-4522-4308-5.
9. Young (2022), p. 16.
10. Bogachev, Vladimir; Smolyanov, Oleg (2020). Real and Functional Analysis. Springer. p. 490. doi:10.1007/978-3-030-38219-3. ISBN 978-3-030-38219-3.
11. Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007). Calculus (9th ed.). Pearson Prentice Hall. p. 332. ISBN 978-0131469686.
12. Hass, Joel R.; Heil, Christopher E.; Weir, Maurice D.; Thomas, George B. (2018). Thomas' Calculus (14 ed.). Pearson. pp. 7–8. ISBN 9780134439020.
13. Anton, Howard; Bivens, Irl; Davis, Stephen (2012). Calculus: Early Transcendentals (9th ed.). John Wiley & Sons. p. 28. ISBN 9780470647691.
14. Varberg, Purcell & Rigdon (2007), pp. 32–33.