Problem 42 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \sqrt{-12}(\sqrt{-4}-\sqrt{2}) $$

Step-by-Step Solution

Answer

The result in standard form is \(-4\sqrt{3} - 2i\sqrt{6}\).

Step 1: Simplify the square roots

Replace each square root with its simplified form. For negative numbers, the square root is equivalent to the square root of the corresponding positive number, multiplied by \(i\). Thus, \(\sqrt{-12} = 2i\sqrt{3}\) and \(\sqrt{-4} = 2i\). Since \(\sqrt{2}\) is not a negative square root, it can remain the same.

Step 2: Perform the subtraction operation

Now, calculate the subtraction operation in the bracket. Since \(\sqrt{-4}\) is equivalent to \(2i\), and we are subtracting \(\sqrt{2}\), we have \(2i - \sqrt{2}\).

Step 3: Perform the multiplication operation

Multiply \(\sqrt{-12}\) now simplified to \(2i\sqrt{3}\) by the result from Step 2, \(2i - \sqrt{2}\). Remember to apply the distributive law when performing the operation. So, \(2i\sqrt{3} * (2i - \sqrt{2}) = 4i^2\sqrt{3} - 2i\sqrt{6}\). Since \(i^2 = -1\), this simplifies further to \(-4\sqrt{3} - 2i\sqrt{6}\).