Problem 49 - Algebra 1

Question

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$\frac{7 \pm 3 \sqrt{2}}{-1}$$

Step-by-Step Solution

Answer

The final answers are -10.24 and -3.76.

Step 1: Find the square root of 2

Using a calculator, find the square root of 2. The square root of 2 is approximately 1.41.

Step 2: Plug square root into the equation

Plug the value obtained into the equation. This will give two expressions: \(\frac{7 + 3*1.41}{-1}\) and \(\frac{7 - 3*1.41}{-1}\).

Step 3: Evaluate the two expressions

Evaluate the expressions. The first expression equals approximately -10.24 and the second expression equals to approximately -3.76.

Step 4: Round to the nearest hundredth

Round the results from the last step to the nearest hundredth. So the final answers are -10.24 and -3.76.

Key Concepts

Square Root

The concept of the square root is fundamental in many areas of mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. In our exercise, we need to find the square root of 2.

Using a calculator, we find that the square root of 2 is approximately 1.41. Here's how it works:

• If you multiply 1.41 by itself (1.41 * 1.41), you get a value close to 2.
• Because the square root is often irrational, we rely on approximations like 1.41 for manual calculations.
• This approximation brings us close to the actual value in practical scenarios.

Understanding square roots helps in solving quadratic equations and other algebraic expressions that involve roots. It is essential not only in math classes but also in science and engineering when dealing with phenomena such as waves and electricity.

Rounding Numbers

Rounding numbers is an essential skill, often used to make numbers easier to work with. It's particularly useful when precision is less critical, allowing for simplified estimates. In this exercise, rounding is employed after calculation to provide a cleaner result.

To round numbers to the nearest hundredth:

• Look at the number in the thousandths place, which is the third digit after the decimal point.
• If this digit is 5 or greater, increase the digit in the hundredths place by one.
• If this digit is less than 5, leave the digit in the hundredths place as it is.

In our example, the results of the calculations were approximately -10.243 and -3.756.

These are rounded to -10.24 and -3.76, making them easier to read and use in further calculations or reports.

Calculator Use

Calculators are invaluable tools in performing complex arithmetic quickly and accurately. They are especially helpful in evaluating algebraic expressions like the one in our exercise. Even though basic operations can be done by hand, calculators streamline the process for more complicated tasks.

When using a calculator:

• Always enter expressions carefully to avoid mistakes.
• Follow the order of operations (first parentheses, then exponents, followed by multiplication and division, and finally addition and subtraction).
• For square roots, use the square root function instead of manually estimating the root.

The efficiency of calculators helps to focus on understanding the concept rather than getting bogged down by arithmetic. In the context of our exercise, the calculator helped us find accurate decimal approximations efficiently, which is essential when preparing results for rounding.