Problem 10 - Elementary Algebra

Question

Simplify each by performing the indicated operation. $$ \frac{4+\sqrt{5}}{3-\sqrt{8}} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(\frac{12+4\sqrt{8}+3\sqrt{5}+\sqrt{40}}{14}\).

Step 1: Identify the conjugate of the denominator

First, we need to determine the conjugate of the denominator. The denominator is \(3-\sqrt{8}\). Its conjugate will be \(3+\sqrt{8}\), which is obtained by changing the sign of the square root term.

Step 2: Multiply the numerator and the denominator by the conjugate

Next, we will multiply both the numerator and the denominator of the given expression by the conjugate we found in Step 1: $$ \frac{4+\sqrt{5}}{3-\sqrt{8}}\cdot\frac{3+\sqrt{8}}{3+\sqrt{8}} $$

Step 3: Expand the numerator and the denominator

Now, we will expand the numerator and the denominator by applying the distributive property. Numerator: $$ (4+\sqrt{5})(3+\sqrt{8})=4\cdot3+4\cdot\sqrt{8}+\sqrt{5}\cdot3+\sqrt{5}\cdot\sqrt{8}=12+4\sqrt{8}+3\sqrt{5}+\sqrt{40} $$ Denominator: $$ (3-\sqrt{8})(3+\sqrt{8})=3\cdot3+3\cdot\sqrt{8}-\sqrt{8}\cdot3-\sqrt{8}\cdot\sqrt{8}=9+5=14 $$

Step 4: Write the simplified expression

Now that we have expanded the numerator and the denominator, we can write the simplified expression: $$ \frac{12+4\sqrt{8}+3\sqrt{5}+\sqrt{40}}{14} $$ So, the simplified expression after performing the indicated operation is: $$ \frac{12+4\sqrt{8}+3\sqrt{5}+\sqrt{40}}{14} $$

Key Concepts

Conjugate

When dealing with expressions that have radicals in the denominator, rationalizing the denominator is a common practice. This means transforming the denominator into a rational number—one without any radicals. To accomplish this, you need to know about conjugates. A conjugate is formed by taking a binomial expression and changing the sign between its terms. For example, if you have a denominator like \(3 - \sqrt{8}\), its conjugate would be \(3 + \sqrt{8}\).

Here are a few reasons to work with conjugates:

• Multiplying by the conjugate can help eliminate radicals in the denominator.
• It's a key technique for simplifying expressions involving radicals.
• Using conjugates helps in achieving a standard form for expressions, making them easier to work with and understand.

By utilizing the conjugate, students simplify complex fractions, making them easier to evaluate or further simplify.

Distributive Property

The distributive property is a fundamental property of numbers, stating that multiplication over a sum or difference can be "distributed" to each term inside. This property is depicted as: \(a(b + c)= ab + ac\). It allows for the distribution of multiplication across terms in a binomial. In our example of \((4+\sqrt{5})(3+\sqrt{8})\), the distributive property expands the expression as follows:

- Multiply each term in the first binomial by each term in the second:

• \(4 \cdot 3\) results in \(12\)
• \(4 \cdot \sqrt{8}\) gives \(4\sqrt{8}\)
• \(\sqrt{5} \cdot 3\) results in \(3\sqrt{5}\)
• \(\sqrt{5} \cdot \sqrt{8}\) gives \(\sqrt{40}\)

Using the distributive property, all terms are combined to form the full expression from each individual multiplication. Manipulating expressions in this way is essential for simplifying and working with algebraic equations and is integral to solving problems with radicals.

Simplification of Radical Expressions

Simplifying radical expressions involves manipulating the expression into its simplest form. Two primary focuses are eliminating radicals from denominators and combining like terms. After utilizing the conjugate and expanding using the distributive property, you support expression simplification by:

1. **Rationalizing the denominator**: After multiplying by the conjugate, the radical in the denominator should ideally disappear, resulting in a simple number as shown in our problem where \((3 - \sqrt{8})(3 + \sqrt{8}) = 14\).

2. **Simplifying the numerator**: Combine like radical terms. For instance, in the numerator \(12 + 4\sqrt{8} + 3\sqrt{5} + \sqrt{40}\), simplify radicals where possible, such as \(\sqrt{8} = 2\sqrt{2}\) and \(\sqrt{40} = 2\sqrt{10}\), to combine like terms.

3. **Final expression simplification**: Write the expression in the simplest form by ensuring no radicals are left in the denominator, and all like radical terms are combined.

These steps ensure a neat, manageable form, aiding further calculations or evaluations while enhancing understanding of the structure of radical expressions.