الأُسُّ في الحساب Power ناتج عدد مضروب في نفسه عددًا محدَّدًا من المرَّات. على سبيل المثال ثلاثة × ثلاثة × ثلاثة × ثلاثة × ثلاثة يسمى الأس الخامس للعدد ثلاثة ويخط ¹3. وفيما يتعلق بـ ¹3 يسمى العدد الأساس والعددخمسة الدليل الأسي. أما الأُسَّين الثاني والثالث لعدد ما فيسميان التربيع ؛ التكعيب . والأس الأول لعدد ما العدد ذاته، أما أس الصفر لعدد ما فهوواحد، بمعنى ثلاثة ¥هوثلاثة و3¤قد يكون 1. وينطبق مفهوم الأس أيضًا على الأعداد والكسور السالبة
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:
From the associativity of multiplication, it follows that for any positive integers m and n,
Any nonzero number raised to the 0 power is 1:
One interpretation of such a power is as an empty product.
The case of 00 is more complicated, and the choice of whether to assign it a value and what value to assign may depend on context. نطقب:Crossref
The following identity holds for an arbitrary integer n and nonzero b:
Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity (∞).
The identity above may be derived through a definition aimed at extending the range of exponents to negative integers.
For non-zero b and positive n, the recurrence relation above can be rewritten as
By defining this relation as valid for all integer n and nonzero b, it follows that
and more generally for any nonzero b and any nonnegative integer n,
This is then readily shown to be true for every integer n.
Identities and properties
The following identities hold for all integer exponents, provided that the base is non-zero:
Unlike addition and multiplication:
- Exponentiation is not commutative. For example, 23 =ثمانية ≠ 32 = 9.
- Exponentiation is not associative. For example, (23)4 = 84 = 4096, whereas 2(34) = 281 = 4178516392292583494123522. Without parentheses, the conventional order of operations in superscript notation is top-down (or right-associative), not bottom-up (or left-associative). That is,
which, in general, is different from
For nonnegative integers n and m, the value of nm is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet). Some examples for particular values of m and n are given in the following table:
nm The nm possible m-tuples of elements from the set نطقب:Mset none
List of whole-number powers
An nth root of a number b is a number x such that xn = b.
If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal th root of b. It is denoted n√, where √ is the radical symbol; alternatively, the principal root may be written b1/n. For example: 91/2 = √ = 3 and 81/3 = 3√ = 2.
The fact that solves follows from noting that
If n is even and b is positive, then xn = b has two real solutions, which are the positive and negative nth roots of b, that is, b1/n > 0 and −(b1/n) < 0.
If n is even and b is negative, the equation has no solution in real numbers.
If n is odd, then xn = b has exactly one real solution, which is positive if b is positive (b1/n > 0) and negative if b is negative (b1/n < 0).
Taking a positive real number b to a rational exponent u/v, where u is an integer and v is a positive integer, and considering principal roots only, yields
Taking a negative real number b to a rational power u/v, where u/v is in lowest terms, yields a positive real result if u is even, and hence v is odd, because then bu is positive; and yields a negative real result, if u and v are both odd, because then bu is negative. The case of even v (and, hence, odd u) cannot be treated this way within the reals, since there is no real number x such that x2k = −1, the value of bu/v in this case must use the imaginary unit i, as described more fully in the section § Powers of complex numbers.
Thus we have (−27)1/3 = −3 and (−27)2/3 = 9. The number أربعة has two 3/2 powers, namelyثمانية and −8; however, by convention the notation 43/2 employs the principal root, and results in 8. For employing the v-th root the u/v-th power is also called the v/u-th root, and for even v the term principal root denotes also the positive result.
This sign ambiguity needs to be taken care of when applying the power identities. For instance:
is clearly wrong. The problem starts already in the first equality by introducing a standard notation for an inherently ambiguous situation –asking for an even root– and simply relying wrongly on only one, the conventional or principal interpretation. The same problem occurs also with an inappropriately introduced surd-notation, inherently enforcing a positive result:
In general the same sort of problems occur for complex numbers as described in the section § Failure of power and logarithm identities.
- Double exponential function
- Exponential decay
- Exponential growth
- List of exponential topics
- Modular exponentiation
- Scientific notation
- Unicode subscripts and superscripts
- Zero to the power of zero
- ^ Achatz, Thomas (2005). (3rd ed.). Industrial Press. p. 101. ISBN .
- ^ Raphael M. Robinson (1958). + 1 and on factors of Fermat numbers" (PDF). Proc. Amer. Math. Soc. 9 (5): 677. doi:10.1090/s0002-9939-1958-0096614-7.
- Introducing 0th power على بلانيت ماث
- Laws of Exponents with derivation and examples