Problem 77 - Algebra for College Students

Question

Does \(\sqrt{x+y}\) equal \(\sqrt{x}+\sqrt{y}\) ? Defend your answer.

Step-by-Step Solution

Answer

No, it's false; \(\sqrt{x+y}\) does not equal \(\sqrt{x} + \sqrt{y}\).

Step 1: Understand the Question

The question is asking if the square root of the sum of two numbers, \(x + y\), is the same as the sum of the square roots of each number, \(\sqrt{x} + \sqrt{y}\).

Step 2: Consider a Counterexample

One way to determine if the statement is true or not is by testing specific values for \(x\) and \(y\). For instance, let's choose \(x = 4\) and \(y = 9\).

Step 3: Calculate \(\sqrt{x+y}\)

First, find the value of \(\sqrt{x+y}\). With \(x = 4\) and \(y = 9\), we have:\[x + y = 4 + 9 = 13\]So:\[\sqrt{x+y} = \sqrt{13}\approx 3.605\]

Step 4: Calculate \(\sqrt{x} + \sqrt{y}\)

Next, calculate \(\sqrt{x} + \sqrt{y}\):\[\sqrt{x} = \sqrt{4} = 2\]\[\sqrt{y} = \sqrt{9} = 3\]Thus:\[\sqrt{x} + \sqrt{y} = 2 + 3 = 5\]

Step 5: Compare Results

Now compare the two results, \(\sqrt{x+y}\) and \(\sqrt{x} + \sqrt{y}\):\[\sqrt{x+y} = \sqrt{13} \approx 3.605\]\[\sqrt{x} + \sqrt{y} = 5\]The two results are different, indicating the original statement is false.

Step 6: Conclusion

By testing the specific values, we observe that \(\sqrt{x+y} eq \sqrt{x} + \sqrt{y}\) in general. Hence, the statement is false.

Key Concepts

Counterexample

A counterexample is a powerful tool in mathematics used to prove that a general statement is false. It identifies one single example where the statement does not hold true.
By finding such an example, we demonstrate that no matter how many cases support the statement, it cannot be labeled as universally true.
In the given exercise, a counterexample was used to determine the truth of the statement \(\sqrt{x+y} = \sqrt{x} + \sqrt{y}\).

• Choosing specific numbers \(x = 4\) and \(y = 9\) helped us investigate the claim.
• We calculated both sides of the equation separately and found they were not equal: \(\sqrt{13} \approx 3.605\) and \(5\).

This counterexample effectively indicates the statement is false as it fails in this instance. This approach is both simple and concrete, making it invaluable for quickly answering similar ambiguities in mathematical inquiries.

Inequality in Mathematics

Inequality plays a central role in understanding mathematical relations.
It explores how different quantities relate to one another when they are not precisely equal.
The concept is fundamental as it includes symbols like \(\leq, \geq, >, <,\), which express less than, greater than, and other relations.

• In the exercise, \(\sqrt{x+y}\) was found to be less than \(\sqrt{x} + \sqrt{y}\), thus showcasing an instance of inequality.
• This illustrates that, in general, the square root of a sum is usually not equal to the sum of square roots, highlighting potential disparities.

Understanding inequalities is crucial for solving problems as it guides us in creating exact mathematical models that reflect real-world situations.

Mathematical Proofs

Mathematical proofs provide a rigorous foundation for verifying the truth of mathematical statements.
They are systematic processes that explain why a statement is necessarily true or false.
There are various types of proofs, such as direct, indirect, and proof by contradiction, each serving unique purposes in mathematics.

• While the exercise used a counterexample to exhibit the statement as false, typically, a formal proof would follow.
• These proofs might involve demonstrating properties of operations and functions to establish conclusive evidence.

Though not every problem requires a full-blown proof, understanding this concept fosters logical thinking and precise mathematical communication, ensuring a deeper grasp of mathematics.