Problem 131 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I simplified the terms of \(2 \sqrt{20}+4 \sqrt{75},\) and then I was able to add the like radicals.

Step-by-Step Solution

Answer

The statement does not make sense because the simplified version of the original expression, \(4\sqrt{5} + 20\sqrt{3}\), does not contain like radicals that can be added together.

Step 1: Simplify the terms separately

Simplify the square roots in the expression \(2 \sqrt{20}\) and \(4 \sqrt{75}\) separately. The square root of 20 can be broken down into \(\sqrt{4} \times \sqrt{5}\), giving \(2 \sqrt{5}\). So, \(2 \sqrt{20}\) becomes 4\( \sqrt{5}\). Similarly, the square root of 75 can be broken down into \(\sqrt{25} \times \sqrt{3}\), giving \(5 \sqrt{3}\). So, \(4 \sqrt{75}\) becomes 20\( \sqrt{3}\).

Step 2: Add the results of step 1

Now add 4\( \sqrt{5}\) and 20\( \sqrt{3}\) together. The result is \(4 \sqrt{5} + 20 \sqrt{3}\). As these are not like terms (since one is the multiple of \(\sqrt{5}\) and the other is the multiple of \(\sqrt{3}\)), they cannot be combined.

Step 3: Assess if the statement makes sense

The original statement said 'I simplified the terms of \(2 \sqrt{20}+4 \sqrt{75}\), and then I was able to add the like radicals'. However, the simplified version of the original expression, \(4 \sqrt{5} + 20 \sqrt{3}\), does not contain 'like radicals' to add. Therefore, the statement does not make sense.