Chapter A, Problem 43E

To determine

To calculate: To prove that $\left|ab\right|=\left|a\right|\left|b\right|$

Answer to Problem 43E

Is proved that $\left|ab\right|=\left|a\right|\left|b\right|$

Explanation of Solution

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

Formula Used:

If $\left|x\right|=a$ , then $x=\pm a$ and $x=\sqrt{a^2}$

Calculation:

Equation is given as

  $\left|ab\right|=\left|a\right|\left|b\right|$

Consider left hand side of the equation,

  $\begin{array}{l}\left|ab\right|=\sqrt{a^2{b}^2}\\ \left|ab\right|=\sqrt{a^2}\times \sqrt{b^2}\\ \left|ab\right|=\left|a\right|\times \left|b\right|\\ \left|ab\right|=\left|a\right|\left|b\right|\end{array}$

Thus, left hand side of equation is equal to right hand side of equation

Conclusion:

Hence, proved that $\left|ab\right|=\left|a\right|\left|b\right|$