Chapter 8.3, Problem 74STP

To determine

To Solve: The inequality and check the solution.

Answer to Problem 74STP

  $p\ge 2$

Explanation of Solution

Given: ${32}^{5p+2}\ge {16}^{5p}$

Given: ${32}^{5p+2}\ge {16}^{5p}$

Solving

  $\begin{array}{l}{32}^{5p+2}\ge {16}^{5p}\\ 2^{5(5p+2)}\ge 2^{4(5p)}\\ 2^{25p+10}\ge 2^{20p}\\ 25p+10\ge 20p\\ 25p-20p\ge 10\\ 5p\ge 10\\ p\ge 2\end{array}$

Checking

Taking $p=2$

  $\begin{array}{l}{32}^{5p+2}\ge {16}^{5p}\\ {32}^{5(2)+2}\ge {16}^{5(2)}\\ {32}^{12}\ge {16}^{10}\end{array}$

Hence, $p\ge 2$