Problem 26 - Prealgebra

Question

Use a calculator to show that \(\sqrt{5}-\sqrt{2}\) is not the same as \(\sqrt{3}.\)

Step-by-Step Solution

Answer

\(\sqrt{5} - \sqrt{2} \neq \sqrt{3}\) because their values are approximately 0.822 and 1.732, respectively.

Step 1: Calculate \(\sqrt{5}\)

Use your calculator to find the square root of 5. When you input 5 and press the square root button, you should get the value \(\sqrt{5} \approx 2.236\).

Step 2: Calculate \(\sqrt{2}\)

Now, find the square root of 2 using your calculator. Input 2 and press the square root button to get \(\sqrt{2} \approx 1.414\).

Step 3: Subtract \(\sqrt{2}\) from \(\sqrt{5}\)

Subtract the value of \(\sqrt{2}\) from \(\sqrt{5}\):\[2.236 - 1.414 = 0.822\]

Step 4: Calculate \(\sqrt{3}\)

Find the square root of 3 using your calculator. Input 3 and press the square root button to get \(\sqrt{3} \approx 1.732\).

Step 5: Compare the Results

Now compare \(\sqrt{5} - \sqrt{2} \approx 0.822\) with \(\sqrt{3} \approx 1.732\). Since \(0.822 eq 1.732\), the two expressions are not the same.

Key Concepts

Square Roots

Square roots are a fundamental concept in math that involves finding a number that, when multiplied by itself, results in the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In mathematical notation, the square root of a number \(x\) is represented as \(\sqrt{x}\). Calculators often have a button dedicated to taking square roots, which is very helpful in quickly finding them for non-perfect squares like 5 or 2.
In our exercise, we deal with \(\sqrt{5}\) and \(\sqrt{2}\). These numbers are irrational, meaning their decimal form goes on forever without repeating. Using a calculator,

• \(\sqrt{5} \approx 2.236\)
• \(\sqrt{2} \approx 1.414\)

Understanding these approximations helps us make further calculations simple.

Comparisons of Expressions

To compare two mathematical expressions, we often look at their simplified numerical forms. In our problem, we are asked to verify whether \(\sqrt{5} - \sqrt{2}\) equals \(\sqrt{3}\). This involves subtraction after finding each square root.
Let's break this down:

• First, find \(\sqrt{5} \approx 2.236\) and \(\sqrt{2} \approx 1.414\).
• Then, perform the subtraction: \(2.236 - 1.414 = 0.822\).
• Lastly, compare this result with \(\sqrt{3} \approx 1.732\).

Since 0.822 does not equal 1.732, the expressions are not the same. This method shows the importance of numerical comparison in verifying mathematical equivalences or inequalities.

Step-by-step Calculations

Step-by-step calculations are highly beneficial for solving math problems, as they help break down complex tasks into smaller, more manageable parts. Let's see how applying these steps clarified the problem:

• **Step 1:** Calculate \(\sqrt{5}\) and get approximately 2.236.
• **Step 2:** Calculate \(\sqrt{2}\) to get about 1.414.
• **Step 3:** Subtract \(\sqrt{2}\) from \(\sqrt{5}\) to find 0.822.
• **Step 4:** Calculate \(\sqrt{3}\) to compare, which is approximately 1.732.
• **Step 5:** Comparing the expressions shows they are not equal as 0.822 \(eq\) 1.732.

This structured approach not only helps in finding the answer but also enhances comprehension and reduces errors. Breaking a problem into these steps has the added benefit of making it easier to troubleshoot if something goes wrong, as you can pinpoint exactly which step needs adjusting. By using calculators wisely in this process, manual errors are minimized, ensuring accuracy.