Question

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$\frac{5 \pm 6 \sqrt{3}}{3}$$

Short answer

The rounded-off answers are \(5.31\) when using the positive version and \(-5.31\) when using the negative version.

Step-by-step solution

Evaluate with positive square root

First, plug in the numbers into the expression using the positive version of \(6 \sqrt{3}\) to get \(\frac{5 + 6 \sqrt{3}}{3}\). Use a calculator to evaluate this expression.

Evaluate with negative square root

Next, evaluate the expression using the negative version of \(6 \sqrt{3}\) to get \(\frac{5 - 6 \sqrt{3}}{3}\). Use a calculator to evaluate this expression.

Rounding the result

After calculating the values, round them off to the nearest hundredths place, as the question requires the answer to be in this format.

Key concepts

Simplifying Radical Expressions

Understanding how to simplify radical expressions is a foundational skill in algebra. It involves expressing the root of a number in the simplest form possible. The process typically includes finding prime factors of the number under the root and simplifying the expression by taking out pairs of prime factors. For example, to simplify \(\sqrt{12}\), one would find that 12 equals 2 x 2 x 3. Since there is a pair of 2's, the root simplifies to \(2\sqrt{3}\).

When dealing with expressions like \(\frac{5 \pm 6 \sqrt{3}}{3}\), the principle remains the same—first focus on simplifying the radical before addressing the rest of the expression. The term \(6\sqrt{3}\) has already been simplified, so the next step would involve performing the addition or subtraction, and then simplifying the fraction if possible. Below are some tips to keep in mind while simplifying radicals.

• Identify and pair up perfect squares within the radical for extraction.
• Maintain any coefficients (like the 6 in our exercise) attached to the radical—these are not part of the number being rooted.
• After simplifying the radical, perform the indicated operations (addition, subtraction, multiplication, division) around it.

Using a Calculator for Algebra

When it comes to algebraic expressions, particularly those involving square roots, calculators can be invaluable tools. For expressions with radicals, like \(\frac{5 \pm 6 \sqrt{3}}{3}\), a calculator that includes a square root function is very helpful.

Here's how to make the most out of a calculator in algebra:

Accurate Input

• Ensure that you enter the expression correctly, paying close attention to the order of operations.
• Use parentheses to denote numerators and denominators in fractions, as well as to correctly input expressions under the root symbol.

Understanding Calculator Syntax

• Be aware of how your calculator interprets button presses; some may require you to enter the radicand (the number under the root) before pressing the root key.
• Consult your calculator’s manual if you’re unsure about its functions and syntax.

Checking Work

• After you get your result, it’s a good practice to approximate the answer mentally or write it down before making calculations to have an expected ballpark figure.
• Always double-check your entries and results to avoid simple input errors.

Calculators can perform the task quickly, but understanding the process mentally reinforces your algebraic knowledge.

Rounding Decimal Places

Rounding to a certain number of decimal places is a mathematical skill used to simplify numbers while maintaining a level of precision. For instance, in situations where highly precise measurements aren'Tnecessary, rounding can make numbers easier to work with. Here's how to round decimals to the nearest hundredth, as the original exercise requires:

First, identify the hundredth place, which is the second digit to the right of the decimal point. Next, look at the third digit (the thousandth place). If this digit is 5 or above, you increase the second digit by one. If it's below 5, you keep the second digit as is.

Consider the number 0.4567. To round to the nearest hundredth, the second digit is a 5, and the third digit is a 6. Since 6 is greater than 5, the number rounded off to the nearest hundredth would be 0.46. When rounding the results from \(\frac{5 \pm 6 \sqrt{3}}{3}\), an understanding of rounding is critical to get the final answer to the required precision.