Problem 25 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of \(\sqrt{2} + \sqrt{3}\) is approximately 3.146, which is not equal to \(\sqrt{5}\), approximately 2.236.

Step 1: Calculate \(\sqrt{2}\)

Using a calculator, find the square root of 2. The value is approximately 1.414.

Step 2: Calculate \(\sqrt{3}\)

Use a calculator to find the square root of 3. The value is approximately 1.732.

Step 3: Add \(\sqrt{2}\) and \(\sqrt{3}\)

Add the approximate values obtained from the calculator: 1.414 + 1.732 = 3.146.

Step 4: Calculate \(\sqrt{5}\)

Find the square root of 5 using a calculator. The value is approximately 2.236.

Step 5: Compare the Results

Compare the sum 3.146 to the square root of 5, approximately 2.236. Since 3.146 is not equal to 2.236, it shows that \(\sqrt{2} + \sqrt{3} eq \sqrt{5}\).

Key Concepts

calculator usage

Using a calculator is a quick and efficient way to get the square roots of numbers that are not perfect squares, like 2 or 3. Calculators come with a square root function which is often represented by the symbol \(\sqrt{}\). To find the square root of any number:\

\
• Enter the number you wish to find the square root of.
\
• Press the square root button (or use a function key if necessary).
\
• Read the approximate result on the display.
\

\
\Remember that calculators provide approximate decimals for square roots, which is sufficient for comparison and many practical purposes. Using a calculator helps in verifying theoretical results quickly and aids in logical reasoning.

comparing inequalities

When comparing inequalities, it's important to understand the concept of relative values. Let’s delve into the example from the exercise where we compare the expression \(\sqrt{2} + \sqrt{3}\) with \(\sqrt{5}\).\
\First, find each square root separately using a calculator as explained above, and then add the results. For \(\sqrt{2} + \sqrt{3}\), you get approximately 3.146. Next, find \(\sqrt{5}\), which is around 2.236.\

\
• If the value of the sum is greater than that of \(\sqrt{5}\), use the symbol \gt\.
\
• If it were less, you would use \lt\.
\
• If they were equal, use the symbol \equiv\.
\

\
\In this case, since 3.146 is significantly larger than 2.236, it clearly shows that \(\sqrt{2} + \sqrt{3}\) is not equal to \(\sqrt{5}\). This process helps strengthen the understanding of inequalities and is an excellent way to test mathematical theories using real numbers.

basic arithmetic operations

Addition and subtraction are some of the basic arithmetic operations that help us combine or separate quantities. In the context of our exercise, addition is used when we combine the square roots calculated: \(\sqrt{2} + \sqrt{3}\).\
\To perform addition accurately:\

\
• Ensure that both numbers you are adding have been correctly calculated or obtained.
\
• Write the numbers in a vertical alignment to help prevent errors when doing it by hand.
\
• Use a calculator for numbers that involve irrational results like square roots to get precise results.
\

\
\The key is to be meticulous and verify each step to avoid mistakes, especially when dealing with irrational numbers. Performing operations such as these builds numerical competency, which is essential in various fields of study.