Problem 41 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-2\sqrt{6} - 2\sqrt{10}i\).

Step 1: Rewrite the square roots of negative numbers

To Rewrite \(\sqrt{-8}\) and \(\sqrt{-3}\), first separate the negative sign from the number and consider it as \(-1\). The square root of \(-1\) is \(i\). This gives \(\sqrt{8}i\) and \(\sqrt{3}i\) respectively. Now let's simplify \(\sqrt{8}\). It can be written as \(\sqrt{4} \times \sqrt{2}\), so \(\sqrt{8}\) simplifies to \(2\sqrt{2}\). Therefore, \(\sqrt{-8}\) can be written as \(2\sqrt{2}i\).

Step 2: Perform the multiplication

Multiply \(2\sqrt{2}i\) with each term inside the parenthesis. So we get \((2\sqrt{2}i \times \sqrt{3}i) - (2\sqrt{2}i \times \sqrt{5})\). Now perform the multiplications \(2\sqrt{2}i \times \sqrt{3}i\) equals \(2\sqrt{6}i^2\) and \(2\sqrt{2}i \times \sqrt{5}\) equals \(2\sqrt{10}i\). Remember that \(i^2 = -1\).

Step 3: Simplify the expression

Now, substitute the value of \(i^2\) in the expression. So it becomes \(2\sqrt{6}(-1) - 2\sqrt{10}i\). By performing the multiplication, the expression is simplified to \(-2\sqrt{6} - 2\sqrt{10}i\).