Chapter 4.3, Problem 50WE

To determine

To simplify the expression

  $5^{2n}\cdot 5^{n+2}\cdot 5^n$

Answer to Problem 50WE

  $5^{4n+2}$

Explanation of Solution

Given:

  $5^{2n}\cdot 5^{n+2}\cdot 5^n$

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 : ${\left(ab\right)}^n=a^n{b}^n$

Calculation:

To simplify the expression

  $5^{2n}\cdot 5^{n+2}\cdot 5^n$

Use the basic properties of power.

First simplify this using the property $a^m{a}^n=a^{m+n}$ and then proceed further to simplify, as

  $\begin{array}{c}{5}^{2n}\cdot 5^{n+2}\cdot 5^n=5^{\left(2n\right)+\left(n+2\right)+\left(n\right)}\\ =5^{2n+n+2+n}\\ =5^{4n+2}\end{array}$

Now the above expression can’t be simplified further.

Thus, simplified form of given expression is

  $5^{2n}\cdot 5^{n+2}\cdot 5^n=5^{4n+2}$