Problem 40 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \frac{4+\sqrt{11}}{4-\sqrt{11}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $$ \frac{4 + \sqrt{11}}{4 - \sqrt{11}} $$ Answer: $$ \frac{27 + 8\sqrt{11}}{5} $$

Step 1: Identify the conjugate

The conjugate of a binomial expression is obtained by changing the sign of the second term. In this case, the conjugate of the denominator, 4 - sqrt(11), would be 4 + sqrt(11).

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

Multiply both the numerator and the denominator of the fraction by the conjugate, 4 + sqrt(11), to eliminate the square root from the denominator: $$ \frac{4 + \sqrt{11}}{4 - \sqrt{11}} \times \frac{4 + \sqrt{11}}{4 + \sqrt{11}} $$

Step 3: Multiply the numerators and denominators

Multiply the numerators and the denominators separately. Use the distributive property (FOIL) to expand the expressions: $$ \frac{(4 + \sqrt{11})(4 + \sqrt{11})}{(4 - \sqrt{11})(4 + \sqrt{11})} $$

Step 4: Simplify the numerator and the denominator

Simplify the numerator and the denominator by applying the difference of squares formula: $$ \frac{4^2 + 2 \cdot 4\sqrt{11} + (\sqrt{11})^2}{4^2 - (\sqrt{11})^2} $$

Step 5: Calculate the result

Now, calculate the result by substituting the values: $$ \frac{16 + 8\sqrt{11} + 11}{16 - 11} $$

Step 6: Simplify the fraction

Combine the like terms in the numerator and simplify the denominator: $$ \frac{27 + 8\sqrt{11}}{5} $$ The simplified expression is: $$ \frac{27 + 8\sqrt{11}}{5} $$

Key Concepts

Conjugate Method

The conjugate method is a technique used in algebra to simplify expressions, especially when dealing with radicals. A conjugate is formed by changing the sign between two terms of a binomial. For instance, if you have an expression like \(4 - \sqrt{11}\), its conjugate would be \(4 + \sqrt{11}\). The main goal of the conjugate method is to remove irrational numbers, like square roots, from the denominator of a fraction.

• This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator.
• This helps to create a difference of squares, which leads to easy simplification.

Understanding the conjugate method is crucial because it lays the foundation for rationalizing denominators, which is an essential skill in algebra.

Rationalizing the Denominator

Rationalizing the denominator means converting a fraction with an irrational denominator into an equivalent fraction with a rational denominator. This process simplifies expressions and makes calculations easier. Imagine you want to simplify \( \frac{4+\sqrt{11}}{4-\sqrt{11}} \).

• First, identify the conjugate of the denominator.
• Then, multiply both the numerator and the denominator by this conjugate.

By doing this, you replace the original denominator with a rational number, eliminating any awkward radicals. This process uses the conjugate method, and is especially useful in mathematical contexts where exact numerical results are important.

Distributive Property (FOIL)

The distributive property is fundamental in multiplying binomials. Specifically, the FOIL method is a mnemonic to help remember the steps for multiplying two binomials: First, Outer, Inner, Last.
For instance, multiplying \((4 + \sqrt{11})(4 + \sqrt{11})\) involves:

• First: Multiply the first terms: \(4 \times 4 = 16\)
• Outer: Multiply the outer terms: \(4 \times \sqrt{11} = 4\sqrt{11}\)
• Inner: Multiply the inner terms: \(\sqrt{11} \times 4 = 4\sqrt{11}\)
• Last: Multiply the last terms: \(\sqrt{11} \times \sqrt{11} = 11\)

Then, combine these results: \[16 + 4\sqrt{11} + 4\sqrt{11} + 11 = 16 + 8\sqrt{11} + 11\]. Using FOIL ensures every part is multiplied correctly, helping avoid common multiplication errors.

Difference of Squares Formula

The difference of squares formula is a powerful tool when simplifying expressions involving squares. It's expressed as \((a + b)(a - b) = a^2 - b^2\). In the context of rationalizing denominators, like in our problem, it simplifies multiplication by using the formula directly.
Applying the formula to \((4 - \sqrt{11})(4 + \sqrt{11})\), you get:

• \(a = 4\) and \(b = \sqrt{11}\)
• Substitute into the formula: \(4^2 - (\sqrt{11})^2\)
• This yields \(16 - 11 = 5\)

This transformation ensures the denominator is rationalized, simplifying the original expression into an easily interpretative form.