Question

Simplify; (a) $f^{3}$ (b) $f^{5}$ (c) $j^{12}$ (d) $f^{14}$,

   Simplify; 
(a) $f^{3}$
(b) $f^{5}$
(c) $j^{12}$
(d) $f^{14}$,

Engineering Mathematics

Engineering Mathematics

K. A. Stroud, Dexter… 5th Edition

Chapter 13, Problem 1

Instant Answer

Solved by Expert Mir Afzal

01/03/2022

Step 1

So, we can express $f^3$ as $f^2 \cdot f$. Substituting $f^2 = -1$, we get $-f$. Show more…

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Simplify; (a) $f^{3}$ (b) $f^{5}$ (c) $j^{12}$ (d) $f^{14}$,

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Key Concepts

Group Theory

Group theory is the branch of abstract algebra that studies algebraic structures known as groups, which consist of a set equipped with an operation satisfying properties like closure, associativity, identity, and inverses. It provides the framework for understanding repeated operations, composition, and symmetry in mathematical objects.

Exponentiation in Groups

Exponentiation in groups refers to the process of composing an element with itself repeatedly, analogous to raising a number to a power. This operation abides by specific algebraic rules that sometimes allow for simplification, especially when the element has a finite order.

Order of an Element

The order of an element in a group is the smallest positive integer n such that raising the element to the nth power results in the identity element of the group. This property is key to simplifying expressions with exponents by reducing the exponent modulo the element's order.

Modular Arithmetic in Exponents

Modular arithmetic in the context of group exponentiation involves reducing the exponent by the order of the element, exploiting the cyclical nature of group elements. This method simplifies expressions with high exponents by taking advantage of the periodic repetition due to the group's structure.

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