Problem 40 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \frac{-15-\sqrt{-18}}{33} $$

Step-by-Step Solution

Answer

The result in standard form is \( -\frac{5}{11} - \frac{\sqrt{2}}{11}i \).

Step 1 - Separate the Square Root into Real and Imaginary Parts

The square root of -18 can be written as \( \sqrt{-1*18} \), which simplifies to \( \sqrt{-1} * \sqrt{18} \). Given that \( \sqrt{-1} = i \) and \( \sqrt{18} = 3\sqrt{2} \), it can be concluded that \( \sqrt{-18} = 3i\sqrt{2} \). Thus, the original problem transforms to \( \frac{-15 - 3i\sqrt{2}}{33} \).

Step 2 - Divide Each Part by the Denominator

Next, each part in the numerator is divided by the denominator, \(33\). Doing this, we get \( \frac{-15}{33} - \frac{3i\sqrt{2}}{33} \). Simplifying this, we obtain \( \frac{-5}{11} - \frac{i\sqrt{2}}{11} \).

Step 3 - Write the Result in Standard Form

The final step is to write the response in the standard form for complex numbers, \(a + bi\), where both \(a\) and \(b\) are real numbers. Hence, the final answer is \( -\frac{5}{11} - \frac{\sqrt{2}}{11}i \).