Chapter 3.7, Problem 80PPE

To determine

Whether the statement is always, sometimes, or never true for real numbers $a$ and $b$.

Answer to Problem 80PPE

The statement is always true for real numbers $a$ and $b$.

Explanation of Solution

Given information: $\left|ab\right|=\left|a\right|.\left|b\right|$

There are three cases.

Case 1

When $a$ and $b$ both are positive.

Let $a=3$ and $b=7$

  $\begin{array}{l}\left|\left(3\right)\left(7\right)\right|=\left|21\right|=21\\ \left|3\right|.\left|7\right|=3.7=21\\ \text{which is true}\text{.}\end{array}$

Case 2

When either $a$ or $b$ are negative.

Let $a=-3$ and $b=7$

  $\begin{array}{l}\left|\left(-3\right)\left(7\right)\right|=\left|-21\right|=21\\ \left|-3\right|.\left|7\right|=3.7=21\\ \text{which is true}\text{.}\end{array}$

Case 3

When both $a$ and $b$ are negative.

Let $a=-3$ and $b=-7$

  $\begin{array}{l}\left|\left(-3\right)\left(-7\right)\right|=\left|21\right|=21\\ \left|-3\right|.\left|-7\right|=3.7=21\\ \text{which is true}\text{.}\end{array}$