Problem 76 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$ 2 \pm 5 \sqrt{3} $$

Step-by-Step Solution

Answer

The two values after rounding to the nearest hundredth are approximately 10.39 for the addition operation and -6.39 for the subtraction operation.

Step 1: Compute the Square Root

Calculate the square root of 3. Use the calculator and get the value of \(\sqrt{3}\).

Step 2: Multiply with 5

Multiply the value obtained in step 1 with 5. The new expression will be \(2 ± 5×(\sqrt{3})\).

Step 3: Add and Subtract from 2

Perform addition and subtraction from 2, leading to two results, one for each operation. So we have \(2 + 5×(\sqrt{3})\) and \(2 - 5×(\sqrt{3})\). Calculate these two values.

Step 4: Rounding

Finally, round each of the results from step 3 to the nearest hundredth.

Key Concepts

Rounding Decimals

Understanding how to effectively round decimals is a fundamental skill in mathematics, especially when it comes to reporting numerical answers with a desired precision. When an exercise instructs you to round to the nearest hundredth, you're essentially looking to truncate the number to two decimal places. Here's how it's done:

For example, if your calculator displays a result like 8.376422, rounding to the nearest hundredth would involve looking at the third decimal place, which is 6 in this instance. Since 6 is 5 or greater, you would round up the second decimal place from 7 to 8, giving you 8.38 as the rounded number.
It's crucial to avoid rounding too early in the problem-solving process, as this can lead to significant errors by the end of your calculations. Always perform all arithmetic operations first, then round the final result as the last step.

Performing Arithmetic Operations

Performing arithmetic operations correctly is the cornerstone of solving algebraic expressions. In exercises like evaluating square roots and their subsequent operations, you must adhere to the Order of Operations—also known as PEMDAS/BODMAS—ensuring you follow the proper sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Using our example, after calculating the square root of 3, you multiply this value by 5 before carrying out the addition and subtraction from 2. This methodical approach ensures that each step is performed accurately and logically. Remember: multiplication and division always precede addition and subtraction unless grouped otherwise by parentheses.

Using Calculators for Algebra

Calculators are incredibly useful tools for handling complex algebraic expressions, especially those involving square roots and other irrational numbers. For instance, squaring roots or computing expressions like \(2 \pm 5 \sqrt{3}\) can be cumbersome by hand.

To use a calculator effectively for algebra, follow these tips:

• Become familiar with your calculator's functions and syntax, especially regarding entering square roots and other operations.
• Always check that the calculator is in the right mode for your calculations (e.g., degree vs. radian mode).
• Use parentheses to group parts of the expression correctly, which will help you avoid syntax errors that can lead to incorrect results.

Consequently, a calculator not only helps you arrive at the correct answer but also saves considerable time on algebraic operations, provided you understand how to leverage its capabilities properly.