Problem 39 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \frac{-6-\sqrt{-12}}{48} $$

Step-by-Step Solution

Answer

-1/8 - i\sqrt{3}/24

Step 1: Recognize the square root of a negative number

Observe that the square root of a negative number, \(\sqrt{-12}\), could be rewritten as \(i\sqrt{12}\) because by definition, \(\sqrt{-1}=i\). Thus, the expression becomes \(-6 - i\sqrt{12}/48\).

Step 2: Simplify the square root

The square root \(\sqrt{12}\) can be simplified by finding the perfect square that goes into 12 which results in \(\sqrt{4*3}\) or \(2\sqrt{3}\). Replace this in the equation, so it becomes \(-6 - 2i\sqrt{3}/48\).

Step 3: Divide each term by 48

The last step is performing the division mentioned in the expression. Applying the division to each term we get: \(-6/48 - 2i\sqrt{3}/48\). This simplifies to \(-1/8 - i\sqrt{3}/24\). This is the final expression in standard form.