Problem 80 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to approximate the number. (Round to three decimal places.)\(\frac{-4+\sqrt{12}}{4}\)

Step-by-Step Solution

Answer

-0.134

Step 1: Calculate the square root

The first step involves calculating the square root of 12. This is done using a calculator. The square root of 12 is approximately 3.464

Step 2: Add the numbers in the numerator

After obtaining the square root of 12, add it to -4. When you add -4 to approximately 3.464, you get approximately -0.536

Step 3: Division

The last step is to divide the result obtained in step 2 by 4. When you divide -0.536 by 4, you get -0.134

Step 4: Rounding to three decimal places

Finally, round the result to three decimal places. The result, -0.134, is already rounded to three decimal places, so no further rounding is necessary

Key Concepts

Rational expressions

Rational expressions are fractions that have a polynomial in both the numerator and the denominator. Understanding how to work with them is essential in algebra. Let's break down the exercise into simpler terms:

• Numerator: \(-4 + \sqrt{12}\)
• Denominator: 4

This specific expression, \(\frac{-4+\sqrt{12}}{4}\), requires combining arithmetic operations with a square root, making it a great example to improve your skills in handling complex rational expressions.

Square roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 12 is a bit tricky because it's not a perfect square like 9 or 16.

In this exercise, we approximate \(\sqrt{12}\) using a calculator:

• \(\sqrt{12} \approx 3.464\)

When dealing with square roots, an approximation often helps in simplifying calculations, especially when the result is part of a larger rational expression.

Decimal approximation

Decimal approximation helps us represent numbers in a simpler, more understandable way. When exact values become cumbersome, like in the case of \(\sqrt{12}\), we use decimals. For this exercise:

You calculate and get -0.536 as the result before dividing by 4. After the division, you need to focus on keeping the result accurate to three decimal places. This means curbing any neglect of significant digits, ensuring precision in your final output.

Using a calculator, you easily achieve this as seen by obtaining \(-0.134\). It's crucial in many algebraic expressions and real-life calculations where precision is key.

Arithmetic calculations

Arithmetic calculations involve basic operations like addition, subtraction, multiplication, and division. To solve the given problem, follow these steps:

• Add \(-4 + 3.464\) to get \(-0.536\).
• Divide \(-0.536\) by 4 which results in \(-0.134\).

Attributes like accuracy come into play here. Whether simple or complex, these operations form the backbone of solving the problem, reflecting how basic arithmetic is interwoven with algebra to address more complicated tasks like rational expressions.