Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression 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. $$ \sqrt{5}+\sqrt{\frac{1}{5}} $$

Step-by-Step Solution

Answer

\( \sqrt{5} + \sqrt{\frac{1}{5}} = \frac{6\sqrt{5}}{5} \)

Step 1: Identifying Radicals

We are given the expression \( \sqrt{5} + \sqrt{\frac{1}{5}} \). Notice that the first term is already in its simplest form, while the second term \( \sqrt{\frac{1}{5}} \) is what we need to simplify.

Step 2: Simplifying the Second Radical

To simplify \( \sqrt{\frac{1}{5}} \), we can express it as \( \frac{\sqrt{1}}{\sqrt{5}} \), which simplifies further to \( \frac{1}{\sqrt{5}} \).

Step 3: Rationalizing the Denominator

To remove the radical from the denominator of \( \frac{1}{\sqrt{5}} \), multiply the numerator and denominator by \( \sqrt{5} \). This gives \( \frac{1 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{5}}{5} \).

Step 4: Combining the Terms

Now, combine the two terms: \( \sqrt{5} + \frac{\sqrt{5}}{5} \). Both terms have a common radical part \( \sqrt{5} \).

Step 5: Factoring Out Common Terms

You can factor out \( \sqrt{5} \) from both terms: \( \sqrt{5} \times (1 + \frac{1}{5}) \).

Step 6: Simplifying Inside the Parentheses

Simplify the expression inside the parentheses, \( 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5} \).

Step 7: Expressing the Final Form

The simplified expression becomes \( \sqrt{5} \times \frac{6}{5} = \frac{6\sqrt{5}}{5} \).

Key Concepts

Rationalizing Denominators

Rationalizing denominators is a technique used in algebra to eliminate radicals, such as square roots, from the denominator of a fraction. When dealing with fractions, having a radical in the denominator can make calculations more complicated. Instead, we aim to have a rational number at the bottom. This process is achieved by multiplying both the numerator and the denominator by a suitable form of 1 which involves the radical.

For example, consider the expression \( \frac{1}{\sqrt{5}} \). The denominator \( \sqrt{5} \) is irrational, so to rationalize it, multiply both the top and bottom by \( \sqrt{5} \). This results in the new expression:
\[ \frac{1 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{5}}{5} \]
This transformation removes the radical from the denominator, making the expression much simpler to work with.

• This method ensures a cleaner algebraic expression.
• Makes further algebraic manipulation more straightforward.

Rationalizing help in keeping expressions neat and often leads to easier further simplification of the expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators such as addition, subtraction, multiplication, and division. They can also involve powers and roots, such as in the expression \( \sqrt{5} + \sqrt{\frac{1}{5}} \).

Understanding how to manipulate these expressions is crucial for solving algebraic equations. Simplification is often the first step. To simplify, we need to:
- Recognize like terms and combine them. For example, notice both terms in our expression involve \( \sqrt{5} \).
- Simplify radicals, such as rewriting \( \sqrt{\frac{1}{5}} \) as \( \frac{\sqrt{5}}{5} \) after rationalizing.

• Combining terms involves factoring common parts.
• Use properties of radicals for simplification.

The more we practice these skills, the more efficiently we can solve complex algebraic problems. Always think about the expression's structure and how best to manipulate it for simplification.

Factoring Common Terms

Factoring involves rewriting an expression as a product of its factors. When expressions share common parts, life can become easier if these are factored out.

In our example, once we have simplified \( \sqrt{5} + \frac{\sqrt{5}}{5} \), both terms share the common factor \( \sqrt{5} \). Factoring it out results in:
\( \sqrt{5} \times (1 + \frac{1}{5}) \).

This step is handy because:

• It highlights the commonality in the terms.
• Makes further manipulation and simplification easier.

After factoring, simplify within the parentheses by performing arithmetic operations, such as converting fractions like \( 1 + \frac{1}{5} = \frac{6}{5} \). This approach not only simplifies the expression but also presents it in an elegant, streamlined form, further easing algebraic computations.