Chapter 12.3, Problem 14E

To determine

To simplify: expression ${(\sqrt{-4}\cdot \sqrt{9})}^2$

Answer to Problem 14E

The required answer is: $-36$

Explanation of Solution

Given information:

The given expression: ${(\sqrt{-4}\cdot \sqrt{9})}^2$

Calculation:

  ${(\sqrt{-4}\cdot \sqrt{9})}^2={(\sqrt{-1\cdot 4}\cdot \sqrt{9})}^2$

(Product property of square roots)

  $\begin{array}{}$ ={(\sqrt{-1\cdot 2^2}\cdot \sqrt{3^2})}^2$$ ={(2i\cdot 3)}^2$$ ={(6i)}^2$$ =36i^2$$ =36(-1)$\end{array}$

(Since $i^2={(\sqrt{-1})}^2=-1$ )

  $=-36$

(Simplify)

The square root of -1 is called an imaginary number and is represented by $i$