Problem 39 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \frac{8+\sqrt{3}}{2+\sqrt{6}} $$

Step-by-Step Solution

Answer

Question: Simplify the complex fraction $\frac{8+\sqrt{3}}{2+\sqrt{6}}$. Answer: $-8+4\sqrt{6}-\sqrt{3}+\frac{3}{2}\sqrt{2}$

Step 1: Identify the conjugate of the denominator

The conjugate of a binomial expression (a+b) is the expression with the opposite sign between the terms (a-b). The conjugate of the denominator in this exercise is (2-√6).

Step 2: Multiply the numerator and denominator by the conjugate

To rationalize the denominator, we will multiply the numerator and denominator by the conjugate (2-√6). When multiplying, remember to distribute each term properly. $$ \frac{8+\sqrt{3}}{2+\sqrt{6}}\times\frac{2-\sqrt{6}}{2-\sqrt{6}} $$

Step 3: Distribute the terms in the numerator and denominator

Now, we will distribute the terms in the numerator and denominator. $$ \frac{(8+\sqrt{3})(2-\sqrt{6})}{(2+\sqrt{6})(2-\sqrt{6})} $$ $$ \frac{16-8\sqrt{6}+2\sqrt{3}-\sqrt{18}}{4-2\sqrt{6}+2\sqrt{6}-6} $$

Step 4: Combine like terms and simplify

Combine any like terms in the numerator and denominator, and then simplify. $$ \frac{16-8\sqrt{6}+2\sqrt{3}-3\sqrt{2}}{-2} $$

Step 5: Divide each term by -2

Now, divide each term in the numerator by -2 to simplify the expression further. $$ \frac{-8+4\sqrt{6}-\sqrt{3}+\frac{3}{2}\sqrt{2}}{1} $$

Step 6: Write the final simplified expression

Since there's no further simplification possible, we can write the final simplified expression as: $$ -8+4\sqrt{6}-\sqrt{3}+\frac{3}{2}\sqrt{2} $$

Key Concepts

Conjugates in Algebra

In algebra, a conjugate is a crucial concept, especially when dealing with complex numbers or radical expressions. Conjugates are pairs of expressions that have the same terms but opposite signs in the middle. Consider the binomial expression \( a + b \): its conjugate would be \( a - b \). This property is especially useful when rationalizing denominators that contain radicals.

• When you multiply a binomial expression by its conjugate, the result is a difference of squares, which elegantly eliminates any radicals in the denominator.
• This process helps make expressions neater and easier to work with, converting them into a format that's more standardized for further mathematical operations.

For instance, in the given problem, the expression in the denominator is \( 2 + \sqrt{6} \). Its conjugate is \( 2 - \sqrt{6} \). By multiplying the numerator and the denominator by this conjugate, the troublesome radical in the denominator can be removed, simplifying the overall expression.

Simplifying Radical Expressions

Dealing with radicals can often seem daunting. However, simplifying radical expressions is a systematic process that breaks down radicals to their simplest form.

• Radicals, such as square roots, need to be simplified by removing as many perfect squares as possible from under the radical sign.
• This can involve factoring numbers under the radical into their smallest components and then simplifying where possible.

In the exercise, multiplying the binomials results in expressions like \( \sqrt{18} \), which can be simplified. By recognizing that \( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \), we reduce complexity, leading to a simplified and more manageable form.

Binomial Expressions

Binomial expressions comprise two distinct terms connected by either addition or subtraction, like \( x + y \) or \( a - b \). These expressions are foundational in algebra. Understanding their properties, especially when dealing with products or powers, is essential.

• When you multiply two binomials together, using the distributive property, you deal with each term individually.
• This operation requires careful attention to detail to ensure all components are rightly multiplied and combined.

The engaging result of these multiplications converges into what is commonly known as 'FOIL' in some teachings, an acronym that stands for First, Outer, Inner, Last – representing the sequence in which terms are multiplied. As applied in the given exercise, multiplying \((8+\sqrt{3})(2-\sqrt{6})\) ensures that each term interacts correctly to form the entire solution, reflecting the mindful application of these principles.