Problem 80 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{10-20 \sqrt{3}}{2} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \( 5 - 10 \sqrt{3} \).

Step 1: Identify the Expression

The expression given is \( \frac{10 - 20 \sqrt{3}}{2} \) which is a fraction that we need to simplify.

Step 2: Break Down the Fraction

The numerator of the fraction is \(10 - 20 \sqrt{3}\) and the denominator is \(2\). The goal is to simplify the fraction by performing division separately for each term.

Step 3: Divide Each Term by the Denominator

Divide each term in the numerator by the denominator separately:\[\frac{10}{2} - \frac{20 \sqrt{3}}{2}\]

Step 4: Simplify Each Division

Perform the division for each term. First calculate \( \frac{10}{2} = 5 \). Next calculate \( \frac{20 \sqrt{3}}{2} = 10 \sqrt{3} \).

Step 5: Combine the Simplified Terms

Combine the results from the previous step:\[5 - 10 \sqrt{3}\]

Step 6: Final Simplified Expression

The expression \( \frac{10 - 20 \sqrt{3}}{2} \) simplifies to \( 5 - 10 \sqrt{3} \).

Key Concepts

Numerator

The numerator is a vital component of a fraction. It is located above the line in a fractional expression. In the given problem, the numerator is \(10 - 20 \sqrt{3}\).Understanding the numerator is crucial when simplifying fractions. It represents the total parts considered from a whole. In more intricate expressions, like our example, the numerator can include constants and expressions involving radicals. Breaking it down step-by-step is essential:

• Consider each part individually when the numerator is made up of multiple terms.
• Apply any operations needed before dividing by the denominator.

In our example, it is essential to separate the constant \(10\) and the radical term \(-20 \sqrt{3}\) before proceeding with the simplification.

Denominator

The denominator in a fraction is found below the fraction bar and indicates the total parts the whole is divided into. In our exercise:\[\text{Denominator} = 2\]The denominator is critical for fraction simplification because the entire numerator is divided by it. When simplifying fractions:

• Always divide each term in the numerator independently by the denominator, as shown in our solution.
• Ensure that the division applies evenly throughout; inaccuracies can alter the entire expression.

By breaking down the numerator into separate terms and dividing each by the denominator, we smoothly simplify the fractional expression.

Radicals

A radical is a mathematical symbol representing the root of a number. In our example, \(\sqrt{3}\) is a radical found in the numerator.Radicals require careful handling during fraction simplification. They often appear in intermediate steps of mathematics problems. Here's how:

• When dividing expressions that include radicals, treat the radical terms independently, such as \(-20 \sqrt{3}\) divided by the denominator.
• Ensure precision during operations to avoid altering the value, as radicals are irrational numbers.

Understanding radicals helps in managing complex fractions efficiently, ensuring they are simplified correctly and accurately.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, making them easier to interpret and use. Our exercise begins with the expression:\(\frac{10 - 20 \sqrt{3}}{2}\).This step-by-step approach:

• First, identify and separate the numerators: \(10\) and \(-20 \sqrt{3}\).
• Then, divide each by the denominator. Simplifying \(\frac{10}{2} = 5\) and \(\frac{20 \sqrt{3}}{2} = 10 \sqrt{3}\).
• Combine the simplified terms to evaluate the final expression: \(5 - 10 \sqrt{3}\).

By following these systematic steps, we ensure effective fraction simplification. It's a logical process that applies to any fractional expression.