Problem 75 - College Algebra with Modeling and Visualization

Question

Use a calculator to evaluate the expression. Round your result to the nearest thousandth. $$ 15+\frac{4+\sqrt{3}}{7} $$

Step-by-Step Solution

Answer

The evaluated expression is approximately 15.819.

Step 1: Simplify Inside the Square Root

First, we look inside the square root symbol. Compute the square root of 3 which will require a calculator. Using the calculator, we find \( \sqrt{3} \approx 1.732 \). Round this to the nearest thousandth.

Step 2: Evaluate the Numerator

Next, simplify the expression in the numerator: \( 4 + \sqrt{3} \). Substitute the approximate value of \( \sqrt{3} \) from Step 1:\[4 + 1.732 = 5.732\]Ensure this value is rounded to the nearest thousandth, which is already done in this case.

Step 3: Divide the Simplified Numerator by 7

Now, divide the result from Step 2 by 7 to simplify the fraction part of the expression:\[\frac{5.732}{7} \approx 0.819\]Use a calculator to perform this division and round the result to the nearest thousandth.

Step 4: Add to 15

Finally, add the result from Step 3 to 15 to complete the evaluation of the original expression:\[15 + 0.819 = 15.819\]This is the final value of the expression rounded to the nearest thousandth.

Key Concepts

Square Roots

The **square root** is an important concept in algebra, helping us figure out which number, when multiplied by itself, yields a given number. In our exercise, we are dealing with the square root of 3, denoted as \( \sqrt{3} \). Quite often, finding the exact square root manually for non-perfect numbers isn't feasible, so we rely on calculators. Here, using a calculator, we get \( \sqrt{3} \approx 1.732 \), a numerical representation rounded to three decimal places.
Understanding square roots involves:

• Observing that for a number \( x \), its square root is a value which, when squared, returns \( x \). For example, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \).
• Using them in expressions where simplification is necessary, as seen in our problem where it's plugged into the numerator.

Square roots are vital in various mathematical and real-world applications due to their utility in calculations involving areas, volumes, and more.

Numerator and Denominator

In algebraic expressions involving fractions, two key components are the **numerator and the denominator**. The numerator is at the top and indicates how many parts we have, whereas the denominator, at the bottom, shows into how many parts the whole is divided.
For our expression \( \frac{4 + \sqrt{3}}{7} \):

• The numerator is \( 4 + \sqrt{3} \). It represents the sum of the integer 4 and the square root value.
• The denominator is 7, showing that we're dividing by 7, thus affecting the whole value derived from the numerator.

In algebra, simplifying the numerator and division by the denominator provides a clearer expression as seen when \( \frac{5.732}{7} \approx 0.819 \) in our problem. Understanding these helps in grasping fractional division and simplification.

Rounding Numbers

**Rounding numbers** is the process of approximating a number to make it simpler and keep it to a desired level of precision. This is essential especially when dealing with irrational numbers or repeating decimals, like the output of dividing fractions including square roots.
In the given problem, the rounding process was used multiple times:

• \( \sqrt{3} \) was rounded to 1.732 to make calculations easier.
• The fraction \( \frac{5.732}{7} \approx 0.819 \) was rounded to the nearest thousandth.
• Finally, the complete expression \( 15 + 0.819 \) results in 15.819, reflecting the rounding to the nearest thousandth.

Rounding numbers allows us to maintain numerical accuracy within a given tolerance, crucial in scientific and economic calculations, as it ensures results remain practical and comprehensible.