Problem 79 - Algebra A Combined Function

Question

Simplify each expression. See Section 10.1. $$ \frac{6+4 \sqrt{5}}{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(3 + 2\sqrt{5}\).

Step 1: Identify Terms in the Numerator

The expression in the numerator is composed of two terms: 6 and \(4\sqrt{5}\). Our goal is to simplify this expression by dividing each term by the denominator.

Step 2: Simplify Each Term

Divide each term in the numerator by the denominator. The expression becomes: \(\frac{6}{2} + \frac{4\sqrt{5}}{2}\).

Step 3: Calculate the Quotient of Each Term

First, calculate \(\frac{6}{2}\), which is 3. Then, calculate \(\frac{4\sqrt{5}}{2}\), which simplifies to \(2\sqrt{5}\).

Step 4: Combine Simplified Terms

Combine the simplified terms from the previous step. The expression is now \(3 + 2\sqrt{5}\).

Key Concepts

Understanding Numerator and Denominator

To solve algebraic fractions like the expression \(\frac{6+4 \sqrt{5}}{2}\), it's crucial to clearly understand what the numerator and denominator are.

- The **numerator** is the top part of the fraction, which in this exercise, consists of two terms: 6 and \(4 \sqrt{5}\).- The **denominator** is the bottom part, which is 2.

The job of simplifying a fraction often involves dividing both the numerator and the denominator by the same number to reduce it to its simplest form.

In this exercise, simplifying means dividing each term of the numerator by the denominator separately, which effectively breaks down the expression into simpler parts.

Simplifying Radicals in Algebraic Expressions

Radicals can seem tricky, but simplifying them is a fundamental part of algebra.
A radical expression includes a radical symbol, often represented as \(\sqrt{}\), which denotes the square root of a number or expression.

When simplifying radicals:

• Look for numbers that can be perfectly squared within the radical.
• If radicals appear in a fraction, they are treated like any other term for simplification purposes.

In our current expression, \(4 \sqrt{5}\) is a radical term. When simplifying the expression, this term becomes \(\frac{4\sqrt{5}}{2}\).
By dividing the coefficient "4" by "2", you get \(2\sqrt{5}\).

The radical part \(\sqrt{5}\) remains unchanged during division, illustrating the importance of understanding how to handle coefficients and radicals separately.

A Step-by-Step Solution Approach

Solving algebraic expressions step by step is a reliable method to ensure complete understanding and accuracy. Let's walk through the solution of the expression \(\frac{6+4 \sqrt{5}}{2}\):

• **Step 1: Identify Terms in the Numerator** - Recognize that the numerator is \(6 + 4 \sqrt{5}\).
• **Step 2: Simplify Each Term** - Divide each term of the numerator by the denominator, 2.
- This transforms the expression to \(\frac{6}{2} + \frac{4\sqrt{5}}{2}\).
• **Step 3: Calculate the Quotient of Each Term** - Compute \(\frac{6}{2}\), which results in 3. - Compute \(\frac{4\sqrt{5}}{2}\), simplifying to \(2\sqrt{5}\).
• **Step 4: Combine Simplified Terms** - Bring together the simplified results to express the final solution as \(3 + 2\sqrt{5}\).

By breaking the process into manageable steps, it becomes much easier to tackle complex expressions and build confidence in algebraic problem-solving.