Problem 50 - Algebra and Trigonometry

Question

Perform the indicated operation(s) and write the result in standard form. $$ 5 \sqrt{-8}+3 \sqrt{-18} $$

Step-by-Step Solution

Answer

The result of performing the indicated operation is \( 19i \sqrt{2} \).

Step 1: Rewrite square roots of negative numbers

Rewrite \( \sqrt{-8} \) and \( \sqrt{-18} \) as \( i \sqrt{8} \) and \( i \sqrt{18} \) respectively, to get the expression \( 5i \sqrt{8} + 3i \sqrt{18} \).

Step 2: Simplify the square roots

Simplify \( \sqrt{8} \) and \( \sqrt{18} \) as \( \sqrt{4} \cdot \sqrt{2} \) and \( \sqrt{9} \cdot \sqrt{2} \) respectively, which further simplifies to \( 2 \sqrt{2} \) and \( 3 \sqrt{2} \). Substitute this to the expression from step 1, to get \( 5i \cdot 2 \sqrt{2} + 3i \cdot 3 \sqrt{2} \).

Step 3: Simplify the multiplication

Perform the multiplication in the expression from step 2 to eventually get \( 10i \sqrt{2} + 9i \sqrt{2} \).

Step 4: Combine like terms

Combine \( 10i \sqrt{2} \) and \( 9i \sqrt{2} \) to get the final result \( 19i \sqrt{2} \).