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Let a = 2 3 3 2 1 4 4 5 6 563 ) and 3 = ( 1 4 2 3 4 5 1 6 3 2 6 5 be permutations.

Let a = 2 3 3 2 1 4 4 5 6 563 ) and 3 = ( 1 4 2 3 4 5 1 6 3 2 6 5 be permutations.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Educational Content on Exponents and Notation** 1. **Find \(\beta^{-1}\). State the answer in array notation.** 2. **Find \(\alpha^{3}\). State the answer in array notation.** 3. **Find \(\beta^{-2}\). State the answer in array notation.** **Understanding the Notation:** - \(\beta^{-1}\) refers to the inverse or reciprocal of \(\beta\). - \(\alpha^{3}\) refers to the cube of \(\alpha\). - \(\beta^{-2}\) refers to the reciprocal of \(\beta\) squared. **Array Notation:** Array notation is a format where elements are arranged in an ordered list or matrix form. In mathematics, expressing the result in array notation means presenting it as an array of values or functions. For example, a vector or a matrix can be considered an array. **Contextual Application:** These expressions are often encountered in algebra and are fundamental in understanding polynomial equations, transformations, and various algebraic manipulations. Each step involves applying exponent rules, such as: - \((x^{-n} = \frac{1}{x^n})\) - \((x^n \cdot x^m = x^{n+m})\) By mastering these expressions and their array notations, students can develop a deeper understanding of algebraic structures and their applications.
Transcribed Image Text:**Educational Content on Exponents and Notation** 1. **Find \(\beta^{-1}\). State the answer in array notation.** 2. **Find \(\alpha^{3}\). State the answer in array notation.** 3. **Find \(\beta^{-2}\). State the answer in array notation.** **Understanding the Notation:** - \(\beta^{-1}\) refers to the inverse or reciprocal of \(\beta\). - \(\alpha^{3}\) refers to the cube of \(\alpha\). - \(\beta^{-2}\) refers to the reciprocal of \(\beta\) squared. **Array Notation:** Array notation is a format where elements are arranged in an ordered list or matrix form. In mathematics, expressing the result in array notation means presenting it as an array of values or functions. For example, a vector or a matrix can be considered an array. **Contextual Application:** These expressions are often encountered in algebra and are fundamental in understanding polynomial equations, transformations, and various algebraic manipulations. Each step involves applying exponent rules, such as: - \((x^{-n} = \frac{1}{x^n})\) - \((x^n \cdot x^m = x^{n+m})\) By mastering these expressions and their array notations, students can develop a deeper understanding of algebraic structures and their applications.
Let \( \alpha = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 4 & 5 & 6 & 3 \end{pmatrix} \) and \( \beta = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 4 & 1 & 3 & 2 & 6 & 5 \end{pmatrix} \) be permutations.
Transcribed Image Text:Let \( \alpha = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 4 & 5 & 6 & 3 \end{pmatrix} \) and \( \beta = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 4 & 1 & 3 & 2 & 6 & 5 \end{pmatrix} \) be permutations.
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