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03.01.2021
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Enroll. Question 3: Consider the following statements: A. Among Sachin, Saurav mathematics equations and formulas pdf 2020 Viru, who received the minimum amount? SBI PO free mock test. In these questions, two equations numbered I and II are given. You have to solve both the equations and select the appropriate option.

In the following questions two equations numbered I and II are given. From statement 1 alone, we do not know if Saurav or Viru has got less than Sachin, from statement 2 alone we cannot know if Sachin or Viru got less than Saurav.

From statement 1, we know mathematics equations and formulas pdf 2020 Sachin received quantities. From 2, we know that Saurav received quantities. Therefore, from both the statements, we know that Sachin received the least amount though we do not know the absolute value of the quantity received, we can mathematics equations and formulas pdf 2020 answer the question.

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To compute i i , write i in polar and Cartesian forms:. The value of a complex power depends on the branch used. The set of all possible values for i i is given by [28].

So there is an infinity of values that are possible candidates for the value of i i , one for each integer k. All of them have a zero imaginary part, so one can say i i has an infinity of valid real values. Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions.

For example:. Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said if only using this result is that:. This identity does not hold even when considering log as a multivalued function. Using Log w for the principal value of log w and m , n as any integers the possible values of both sides are:.

On the other hand, when x is an integer, the identities are valid for all nonzero complex numbers. Exponentiation with integer exponents can be defined in any multiplicative monoid. Exponentiation is defined inductively by. Monoids include many structures of importance in mathematics, including groups and rings under multiplication , with more specific examples of the latter being matrix rings and fields.

If A is a square matrix, then the product of A with itself n times is called the matrix power. Matrix powers appear often in the context of discrete dynamical systems , where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system. So computing matrix powers is equivalent to solving the evolution of the dynamical system.

In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors. Apart from matrices, more general linear operators can also be exponentiated.

The n -th power of the differentiation operator is the n -th derivative:. These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral , is one of the basic operations of the fractional calculus.

A field is an algebraic structure in which multiplication, addition, subtraction, and division are all well-defined and satisfy their familiar properties. The real numbers, for example, form a field, as do the complex numbers and rational numbers.

Unlike these familiar examples of fields, which are all infinite sets , some fields have only finitely many elements. Exponentiation in finite fields has applications in public key cryptography. For example, the Diffie�Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm the inverse of exponentiation is computationally expensive.

This prime number is called the characteristic of the field. This is called the Frobenius automorphism of F. The Frobenius automorphism is important in number theory because it generates the Galois group of F over its prime subfield. Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.

Let X be a set with a power-associative binary operation which is written multiplicatively. Then x n is defined for any element x of X and any nonzero natural number n as the product of n copies of x , which is recursively defined by.

If the operation has a Mathematics Simultaneous Equations Pdf Version two-sided identity element 1, then x 0 is defined to be equal to 1 for any x : [ citation needed ]. If the operation also has two-sided inverses and is associative, then the magma is a group. If the multiplication operation is commutative as, for instance, in abelian groups , then the following holds:. If the binary operation is written additively, as it often is for abelian groups , then "exponentiation is repeated multiplication" can be reinterpreted as " multiplication is repeated addition ".

Thus, each of the laws of exponentiation above has an analogue among laws of multiplication. When there are several power-associative binary operations defined on a set, any of which might be iterated, it is common to indicate which operation is being repeated by placing its symbol in the superscript.

Superscript notation is also used, especially in group theory , to indicate conjugation. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense.

A quandle is an algebraic structure in which these laws of conjugation play a central role. If n is a natural number, and A is an arbitrary set, then the expression A n is often used to denote the set of ordered n -tuples of elements of A.

This generalized exponential can also be defined for operations on sets or for sets with extra structure. For example, in linear algebra , it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of. We can again replace the set N with a cardinal number n to get V n , although without choosing a specific standard set with cardinality n , this is defined only up to isomorphism.

Taking V to be the field R of real numbers thought of as a vector space over itself and n to be some natural number , we get the vector space that is most commonly studied in linear algebra, the real vector space R n. If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian product unless otherwise stated. Since multiple Cartesian products produce an n - tuple , which can be represented by a function on a set of appropriate cardinality, S N becomes simply the set of all functions from N to S in this case:.

In a Cartesian closed category , the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets.

If 0 is an initial object in a Cartesian closed category, then the exponential object 0 0 is isomorphic to any terminal object 1. In set theory , there are exponential operations for cardinal and ordinal numbers.

Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction. Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation.

This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 0 0.

The limits in these examples exist, but have different values, showing that the two-variable function x y has no limit at the point 0, 0. One may consider at what points this function does have a limit. These powers are obtained by taking limits of x y for positive values of x. On the other hand, when n is an integer, the power x n is already meaningful for all values of x , including negative ones. Compute the following in order:.

This series of steps only requires 8 multiplication operations the last product above takes 2 multiplications instead of Finding the minimal sequence of multiplications the minimal-length addition chain for the exponent for b n is a difficult problem, for which no efficient algorithms are currently known see Subset sum problem , but many reasonably efficient heuristic algorithms are available.

Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated function composition rather than repeated multiplication. A similar convention exists for logarithms, [40] where today log 2 x usually means log x 2 , not log log x. For the same purpose, f [ n ] x was used by Benjamin Peirce [55] [40] whereas Alfred Pringsheim and Jules Molk suggested n f x instead.

Programming languages generally express exponentiation either as an infix operator or as a prefix function, as they are linear notations which do not support superscripts:. Many other programming languages lack syntactic support for exponentiation, but provide library functions:. For certain exponents there are special ways to compute x y much faster than through generic exponentiation. Not all programming languages adhere to the same association convention for exponentiation: while the Wolfram language , Google Search and others use right-association i.

From Wikipedia, the free encyclopedia. Mathematical operation. For other uses, see Exponent disambiguation. Arithmetic operations v t e. Exponentiation is not commutative. Exponentiation is not associative. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down or right -associative , not bottom-up [19] [20] [21] [22] or left -associative.