Question

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

Short answer

The expression evaluates to either 2.13 or -1.33

Step-by-step solution

Calculate Square Root

To begin, evaluate the square root part of the expression. The square root of 3 is approximately 1.732.

Perform Addition and Subtraction

Next, calculate the results for the numerator in two cases: one using \(+\) where \(2 + 5 \times 1.732 = 10.66\) and another using \(-\) where \(2 - 5 \times 1.732 = -6.66\).

Divide by Denominator

Now, for each case, divide the solutions found at the numerator by the denominator. In the case of \(+\), \(10.66 \div 5 = 2.13\) and in the case of \(-\), \(-6.66 \div 5 = -1.33\)

Round to Nearest Hundredth

Lastly, round the final answer to the nearest hundredth. The two possible solutions for this expression would be 2.13 (for the \(+\) case) and -1.33 (for the \(-\)) case.

Key concepts

Calculator Usage in Algebra

When working with algebraic expressions, especially those involving square roots and other more advanced operations, a calculator is an invaluable tool. It speeds up the process and helps prevent manual errors. In evaluating expressions like \(\frac{2 \pm 5 \sqrt{3}}{5}\), the calculator allows for an accurate computation of square roots, which are often irrational numbers.

However, one must be familiar with how to input expressions correctly into the calculator. For square roots, calculators typically have a dedicated button, labeled either \(\sqrt{x}\) or something similar. First, input the number you want to take the square root of (in this case, 3), and then hit the square root button. As square root results can be lengthy decimals, it's crucial to know how many decimal places your calculator displays and whether it rounds up or auto-completes long digits.

To ensure accuracy, check your calculator's settings for floating-point arithmetic or rounding preferences. When performing arithmetic operations with these results, as seen in the expression above, use parentheses to group terms and maintain the correct order of operations; for instance, inputting \(2 + 5 * \sqrt{3}\) as one term and then dividing by 5.

Rounding Decimals

Decimals, especially those resulting from square roots or divisions in algebra, can be quite lengthy. Rounding them to a certain number of decimal places can help simplify the numbers for further calculations or for final answers that are concise and easier to comprehend.

Rounding to the nearest hundredth means keeping two digits after the decimal point. For example, if the calculator gives you a long decimal like 2.132879..., rounding to the nearest hundredth results in 2.13. The third digit after the decimal point, which is 2 in this case, is examined. If this digit is 5 or higher, you round up the second digit by one. If it's less than 5, you leave the second digit as is.

This principle is widely used in algebra to help manage infinite decimals that arise from irrational numbers, like square roots of non-perfect squares. It is important to round correctly depending on the context; in sciences, precision matters, while in other subjects such as financial calculations, it might be standard to round to two decimal places.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are fundamental skills necessary for evaluating algebraic expressions. The order in which these operations are performed is critical, adhering to the mathematical principle known as the order of operations, often memorized by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

In the given expression \(\frac{2 \pm 5 \sqrt{3}}{5}\), we first carry out the square root operation, followed by multiplication, then addition or subtraction in the numerator, and finally, the division.

Furthermore, understanding how these operations interact with positive and negative numbers is crucial. For instance, subtracting a larger number from a smaller one, as in \(2 - 5 \times 1.732\), results in a negative number. Division by a positive number maintains the sign, so it's useful to keep these rules in mind to avoid errors in computing and interpreting the results of algebraic expressions.