Problem 43 - College Algebra

Question

Add or subtract terms whenever possible. $$3 \sqrt{8}-\sqrt{32}+3 \sqrt{72}-\sqrt{75}$$

Step-by-Step Solution

Answer

The simplified expression is \(20\sqrt{2} - 5\sqrt{3}\).

Step 1: Simplify Square Root Terms

We start by simplifying each square root term individually.The first square root \(\sqrt{8}\) can be written as \(2\sqrt{2}\) since 8 = 4×2.The term \(\sqrt{32}\) can be simplified to \(4\sqrt{2}\) as 32 = 16×2.Similarly, the term \(\sqrt{72}\) can be simplified to \(6\sqrt{2}\) since 72 = 36×2.Finally, the term \(\sqrt{75}\) can be rewritten as \(5\sqrt{3}\) as 75 = 25×3.

Step 2: Substitute and Perform the Operations

Now substitute \(\sqrt{8}\) by \(2\sqrt{2}\), \(\sqrt{32}\) by \(4\sqrt{2}\), \(\sqrt{72}\) by \(6\sqrt{2}\), and \(\sqrt{75}\) by \(5\sqrt{3}\) in the original expression. You get \(3 \times 2\sqrt{2} - 4\sqrt{2} + 3 \times 6\sqrt{2} - 5\sqrt{3}\) which further simplifies to \(6\sqrt{2} - 4\sqrt{2} + 18\sqrt{2} - 5\sqrt{3}\).

Step 3: Combine Like Terms

The next step is to combine like terms. The terms with \(\sqrt{2}\) are \(6\sqrt{2} - 4\sqrt{2} + 18\sqrt{2}\) and when combined, it gives \(20\sqrt{2}\). So, the final simplified expression is \(20\sqrt{2} - 5\sqrt{3}\).