Let G = (a, b | a² = b² = (ab)?).a. Express b²abab³ in the form b'a'.b. Express b'abab³a in the form b'a'.

Answered: Let G = (a, b | a² = b² = (ab)?). a.…

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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Question

Let G = (a, b | a² = b² = (ab)?).
a. Express b²abab³ in the form b'a'.
b. Express b'abab³a in the form b'a'.

Expert Solution

Step 1

Given: The expression is $G = <a, b | a^{2} = b^{2} = {(a b)}^{2}>$ .

To Express: (a) The expression $b^{2} a b a b^{3}$ in the form of $b^{i} a^{j}$ .

(b) The expression $b^{3} a b a b^{3}$ in the form of $b^{i} a^{j}$ .

Step 2

(a)

Rewrite the given expression,

$b^{2} a b a b^{3} = b^{2} a b a b^{2} b$

Now use the given condition,

$\begin{array}{rcl} b^{2} a b a b^{2} b & = & b^{2} a b a a^{2} b \\ = & b^{2} a b a^{3} b \end{array}$

Again break the exponent,

$\begin{array}{rcl} b^{2} a b a^{3} b & = & b^{2} a b a^{2} a b \end{array}$

Now use the given condition,

$\begin{array}{rcl} b^{2} a b a^{2} a b & = & b^{2} a b b^{2} a b \\ = & b^{2} a b^{3} a b \end{array}$

Again break the exponent,

$\begin{array}{rcl} b^{2} a b^{3} a b & = & b^{2} a b^{2} b a b \end{array}$

Now use the given condition,

$\begin{array}{rcl} b^{2} a b^{2} b a b & = & b^{2} a a^{2} b a b \\ = & b^{2} a^{2} a b a b \\ = & {(a b)}^{2} (a b) (a b) \\ = & {(a b)}^{2} {(a b)}^{2} \\ = & {(a b)}^{4} \\ = & b^{4} a^{4} \end{array}$

Therefore, the given expression can be written as $\begin{array}{rcl} b^{4} a^{4} \end{array}$

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Let G = (a, b | a² = b² = (ab)?). a. Express b²abab³ in the form b'a'. b. Express b'abab³a in the form b'a'.