Problem 35 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \frac{4 \sqrt{3}}{3}-\frac{\sqrt{12}}{3} $$

Step-by-Step Solution

Answer

\( \frac{2 \sqrt{3}}{3} \) is the simplified result.

Step 1: Simplify the Radicals

First, simplify any square roots involved in the expression. We have \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \). Now rewrite the original expression as follows:\[ \frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3} \]

Step 2: Combine Like Terms

Since both terms have the same denominator, \(3\), and the same radical \(\sqrt{3}\), you can combine them using the properties of fractions. Factoring out \(\sqrt{3}\) gives us:\[ \frac{4 \sqrt{3} - 2 \sqrt{3}}{3} = \frac{(4 - 2) \sqrt{3}}{3} \]

Step 3: Perform Subtraction

Subtract the coefficients in the numerator: \(4 - 2 = 2\). So, \[ \frac{(4 - 2) \sqrt{3}}{3} = \frac{2 \sqrt{3}}{3} \]

Step 4: Finalize the Simplified Expression

The simplified expression, after performing the subtraction, results in:\[ \frac{2 \sqrt{3}}{3} \] This is the final expression after simplifying the given radicals.

Key Concepts

Fraction Operations

Fraction operations involve adding, subtracting, multiplying, or dividing fractions. Understanding how to handle these operations helps simplify expressions involving fractions, like the one in our exercise. When you have fractions with the same denominator, as in the problem where both terms have a denominator of 3, you can directly add or subtract the numerators. This makes fraction operations straightforward in such cases, as you only need to focus on the numerators while keeping the denominator constant. Therefore, the task simplifies to:

• Identifying and keeping the common denominator.
• Performing the arithmetic operation (addition or subtraction) on the numerators.

This greatly reduces complexity and helps in quickly solving problems without needing additional steps to find a common denominator.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions by merging terms that have identical variable parts. In the provided example, both of the terms share the same radical, \( \sqrt{3} \), which can be treated as a common part. By treating \( \sqrt{3} \) as a common factor, you essentially reduce the problem to simple arithmetic - working with the numerical coefficients of the like terms: 4 and 2. This process involves:

• Identifying terms that can be combined based on having the same variable or radical part.
• Adding or subtracting their coefficients accordingly.

The combination of like terms simplifies expressions, making them more manageable and helps to focus on the core numerical information.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and mathematical operations such as addition and multiplication. In the given problem, the expression includes radicals and fractions, which are typical components of more complex algebraic expressions. Understanding algebraic expressions requires a few key strategies:

• Simplifying radicals, which helps in breaking down complex expressions into simpler parts.
• Combining like terms to further simplify expressions by reducing the number of unneeded variables or terms.
• Performing operations such as addition, subtraction, multiplication, and division appropriately.

Learning to manipulate algebraic expressions effectively involves practicing these steps until they become intuitive, enabling more efficient and accurate problem-solving.