Problem 24 - Algebra 2 and Trigonometry

Question

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \frac{5+6 \sqrt{5}}{\sqrt{5}} $$

Step-by-Step Solution

Answer

The simplest form is \( \sqrt{5} + 6 \).

Step 1: Break Apart the Fraction

First, break the given fraction \( \frac{5 + 6\sqrt{5}}{\sqrt{5}} \) into two separate fractions: \( \frac{5}{\sqrt{5}} + \frac{6\sqrt{5}}{\sqrt{5}} \). This helps us simplify each part individually.

Step 2: Simplify Each Term

Now simplify each fraction separately. Start with \( \frac{5}{\sqrt{5}} \). You can simplify this as \( \frac{5}{\sqrt{5}} = \frac{5}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{5} = \sqrt{5} \).

Step 3: Simplify Second Term

Now simplify the term \( \frac{6\sqrt{5}}{\sqrt{5}} \). The \( \sqrt{5} \) terms cancel out, leaving \( 6 \).

Step 4: Combine Results

Combine the simplified terms \( \sqrt{5} \) and \( 6 \). Thus, the expression becomes \( \sqrt{5} + 6 \).

Key Concepts

Understanding Fractions

Fractions can sometimes look complicated, especially when they include additional elements like radicals. But, they're not that complex once you break them down. A fraction simply consists of a numerator (top number) and a denominator (bottom number). When you see fractions, remember they represent division: the numerator divided by the denominator. For example, in the expression \( \frac{5 + 6\sqrt{5}}{\sqrt{5}} \), you're essentially dividing one set of quantities by another.When simplifying fractions with additional expressions, you can break down the fraction into simpler parts. This means splitting the expression in the numerator across multiple fractions that share the same denominator. This is what we did in the example above when we created \( \frac{5}{\sqrt{5}} + \frac{6\sqrt{5}}{\sqrt{5}} \). By looking at each part individually, you make the whole expression easier to manage.

Breaking Down Radicals

Radicals can seem tricky, but they're just another way to express square roots or other roots. A radical sign (\( \sqrt{} \)) indicates that you want to find the root of a number or expression. For example, \( \sqrt{5} \) asks for the square root of 5. If you're working with fractional radicals, your goal is often to simplify them.Simplifying radicals means getting them into the simplest form possible. For instance, when you have \( \frac{5}{\sqrt{5}} \), multiplying both numerator and denominator by \( \sqrt{5} \) helps eliminate the radical from the denominator. This clever trick, known as "rationalizing the denominator," transforms \( \frac{5}{\sqrt{5}} \) into \( \sqrt{5} \).For radicals in fractions, dealing with expressions like \( \frac{6\sqrt{5}}{\sqrt{5}} \) requires cancelling out the radicals in both the numerator and denominator (when they are the same), leaving you with a simpler form, such as \( 6 \). Always aim to make radicals as straightforward as possible, adhering to simple arithmetic rules wherever applicable.

Simplifying Algebraic Expressions

Algebraic expressions can include various elements, such as numbers, variables, and radicals. The key to simplifying them is to tackle one component at a time, maintaining simplicity across each term.When working with expressions, remember:

• Break down each component separately. For example, split compounds into simpler fractions.
• Simplify using basic arithmetic operations wherever possible.
• Combine like terms to reach the simplest form.

In the solution \( \sqrt{5} + 6 \), we broke down the original expression into parts and simplified them separately. By taking each fraction apart, rationalizing where necessary, and combining simplified components, we achieved the most reduced form of the algebraic expression.Always follow these steps to break the expression down methodically, ensuring you're finding the simplest form efficiently and accurately.