Problem 12 - Algebra 1: Concepts and Skills

Question

Use a calculator or a table of square roots to evaluate the expression. Round the results to the nearest hundredth. $$ 1 \pm \sqrt{2} $$

Step-by-Step Solution

Answer

The expressions \(1 + \sqrt{2}\) and \(1 - \sqrt{2}\) evaluate to approximately 2.41 and -0.41 respectively when rounded to the nearest hundredth.

Step 1: Find the square root of 2

Using a calculator, find the square root of the number 2. Make sure to round your result to the nearest hundredth to maintain accuracy throughout the problem.

Step 2: Apply the square root to the expression

Now, place the value of \(\sqrt{2}\) found in Step 1 into the original expression, giving you two versions of this expression, one with addition and the other with subtraction.

Step 3: Perform the operations

Perform the addition for the first expression \(1 + \sqrt{2}\) and the subtraction for the second expression \(1 - \sqrt{2}\). Remember to round each answer to the nearest hundredth.

Key Concepts

Rounding Numbers

When solving mathematical expressions, especially those involving irrational numbers like square roots, rounding becomes crucial.
Rounding numbers to the nearest hundredth means you will round your number to two decimal places.
For example, if your number is 1.414213, you look at the third decimal place to decide whether to round up or down. If this number is 5 or more, you round up; if it is 4 or less, you round down.
By rounding to the nearest hundredth, you help ensure your numbers are easier to work with and your calculations are consistent.

• Identify the digit in the place you need to round to.
• Look at the next digit to the right. This will determine if you round up or keep it the same.
• Adjust the number based on the value of the next digit.

Rounding helps maintain simplicity while allowing for a more precise final answer.

Addition and Subtraction of Square Roots

Handling square roots in addition and subtraction can seem tricky, but with practice, it becomes straightforward.
You often encounter problems where you need to add or subtract square roots, like in the expression given, which has two parts: \(1 + \sqrt{2}\) and \(1 - \sqrt{2}\).
The key is to separately evaluate each square root and then perform the addition or subtraction.

• Calculate each square root first to get a decimal value.
• Add or subtract these values as you would with regular numbers.
• Remember: keep your results approximate by rounding to ensure calculations remain manageable.

This process might remind you a bit of how you handle variables, but with every square root simplified to its decimal before doing any operations.

Using a Calculator for Square Roots

Using a calculator is a practical way to find square roots, especially when dealing with non-perfect squares like 2.
Calculators yield an accurate approximation. When using the calculator to find the square root of 2, it provides you with a long decimal, \(\sqrt{2} \approx 1.4142135\).
But since it's not feasible to work with so many decimal places manually, you will need to round your calculation as discussed earlier. Here’s how you can use a calculator for square roots efficiently:

• Enter the number you want the square root of, which is 2 in this case.
• Press the square root function, usually symbolized by \(\sqrt{}\) on the calculator.
• Round the resultant value as needed based on requested precision.

This efficient approach not only saves time but also ensures accuracy, allowing you to focus on further calculations with confidence.