Chapter 4.4, Problem 46WE

To determine

To simplify: the expression

  ${\left(a^x\right)}^y{\left(a^y\right)}^x$

Answer to Problem 46WE

  $a^{2xy}$

Explanation of Solution

Given:

  ${\left(a^x\right)}^y{\left(a^y\right)}^x$

Concept Used:

• For all positive integers m and n : ${\left(a^m\right)}^n=a^{mn}$
• For all integers m and n : $a^m{a}^n=a^{m+n}$
• For all integers m and n : ${\left(ab\right)}^n=a^n{b}^n$
• If an expression contains more than one operations between its terms and brackets, first solve the terms inside brackets and then apply division followed by multiplication, addition and subtraction.

Calculation:

To simplify the expression

  ${\left(a^x\right)}^y{\left(a^y\right)}^x$

Use the basic properties of power.

First simplify this using the property ${\left(ab\right)}^n=a^n{b}^n$ , and then proceed further to simplify, as

  $\begin{array}{c}{\left(a^x\right)}^y{\left(a^y\right)}^x=\left(a^{x\times y}\right)\left(a^{y\times x}\right)\\ =a^{xy}\cdot a^{xy}\\ =a^{xy+xy};\text{ using property }{a}^m{a}^n=a^{m+n}\\ =a^{2xy}\end{array}$

Thus, simplified form of given expression is

  ${\left(a^x\right)}^y{\left(a^y\right)}^x=a^{2xy}$