Problem 44 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ (3 \sqrt{-7})(2 \sqrt{-8}) $$

Step-by-Step Solution

Answer

Hence, the product of the provided expression is \( -6\sqrt{14}\).

Step 1: Convert into Standard Form

First convert the square roots into standard form, separating out any negative roots. \( \sqrt{-7} \) can be written as \( \sqrt{7} \) times \( \sqrt{-1} \). Hence, \( \sqrt{-7} \) is \( i \sqrt{7} \). Similarly, \( \sqrt{-8} \) can be written as \( \sqrt{8} \) times \( \sqrt{-1} \) which simplifies to \( 2i \sqrt{2} \) as \( \sqrt{8} \) simplifies to \( 2\sqrt{2}\).

Step 2: Perform Multiplication

Now we multiply the two values obtained from step 1. So \((3i\sqrt{7})\) times \((2i\sqrt{2})\) equals \(6i^{2}\sqrt{14}\). Since \(i^{2}\) equals -1 (as per the definition of imaginary number i), we substitute \(i^{2}\) with -1.

Step 3: Final Simplification

After the substitution, the expression becomes \( -6\sqrt{14} \). This is the number in standard form.