Problem 30 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \sqrt{-81}-\sqrt{-144} $$

Step-by-Step Solution

Answer

The result of \( \sqrt{-81}-\sqrt{-144} \) in standard form is -3i.

Step 1: Calculate Square Roots

Start by calculating the square roots in isolation, without considering the negative signs. \(\sqrt{-81}\) becomes \(\sqrt{81}\) and \(\sqrt{-144}\) becomes \(\sqrt{144}\). Now, calculate the square roots of these resulting numbers to get 9 and 12.

Step 2: Handle Negative Square Roots

Now, take into account that the original numbers were negative, which imply they are complex numbers. This means that \(\sqrt{-81}\) becomes 9i and \(\sqrt{-144}\) becomes 12i, since the square root of -1 is represented as 'i' in complex numbers.

Step 3: Subtract Complex Numbers

Finally, subtract the second result from the first one, that is 9i - 12i, to get -3i.