Chapter 4.4, Problem 48WE

To determine

To simplify: the expression

  ${\left(2x^n\right)}^3{\left(x^n\right)}^5$

Answer to Problem 48WE

  $8x^{8n}$

Explanation of Solution

Given:

  ${\left(2x^n\right)}^3{\left(x^n\right)}^5$

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

  ${\left(2x^n\right)}^3{\left(x^n\right)}^5$

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

  $\begin{array}{c}{\left(2x^n\right)}^3{\left(x^n\right)}^5=2^3{x}^{n\times 3}\cdot x^{n\times 5}\\ =8x^{3n}\cdot x^{5n}\\ =8x^{3n+5n};\text{ using property }{a}^m{a}^n=a^{m+n}\\ =8x^{8n}\end{array}$

Thus, simplified form of given expression is

  ${\left(2x^n\right)}^3{\left(x^n\right)}^5=8x^{8n}$