Problem 74 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \frac{8}{2-\sqrt{6}} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(-8 - 4\sqrt{6}\).

Step 1: Identify the conjugate of the denominator

The conjugate of a binomial of the form \((a+b)\) is \((a-b)\). So the conjugate of the denominator \((2-\sqrt{6})\) is \((2+\sqrt{6})\).

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

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate: $$ \frac{8}{2-\sqrt{6}} \times \frac{2+\sqrt{6}}{2+\sqrt{6}} $$

Step 3: Perform the operations in the numerator and denominator

Now, multiply the numerators and the denominators respectively: $$ \frac{8 \times (2+\sqrt{6})}{(2-\sqrt{6}) \times (2+\sqrt{6})} $$ Apply the distributive property to the numerator and use the difference of squares property for the denominator: $$ \frac{8(2) + 8\sqrt{6}}{(2)^2 - (\sqrt{6})^2} $$

Step 4: Simplify the expression

Next, simplify the numerator and the denominator: $$ \frac{16 + 8\sqrt{6}}{4 - 6} $$ $$ \frac{16 + 8\sqrt{6}}{-2} $$ Now, divide both the constant and the term with the square root by the constant in the denominator: $$ \frac{16}{-2} + \frac{8\sqrt{6}}{-2} $$ And simplify the fractions: $$ -8 - 4\sqrt{6} $$ So the simplified expression is \(-8 - 4\sqrt{6}\).

Key Concepts

Conjugate

When dealing with radicals in denominators, the conjugate is a powerful tool to simplify them. The conjugate of a binomial expression such as \(a + b\) is obtained by changing the sign between the terms, making it \(a - b\). This strategy is crucial because multiplying a binomial by its conjugate results in a difference of squares.
This approach allows us to remove the square root in the denominator, which makes computations easier and the expression neater.
Remember only to take the conjugate of the denominator when rationalizing; the conjugate of the entire expression may lead to unnecessary complexity. By using the conjugate, you are effectively turning a potentially complicated radical expression into a simpler format.

Difference of Squares

The difference of squares is a simple formula used regularly in algebra to simplify expressions.
This formula states that the product of two conjugates, \((a+b)(a-b)\), results in \(a^2 - b^2\). This principle is essential in rationalizing denominators, especially those involving radicals.
By converting the expression into a difference of squares, square roots cancel out and leave behind easy-to-handle integer terms.

• For example, \((2+\sqrt{6})(2-\sqrt{6}) = 2^2 - (\sqrt{6})^2 = 4 - 6 = -2.\)

Notice how the radical parts disappear, and we're left with a simple subtraction. This dramatic simplification is what makes the difference of squares so valuable.

Simplifying Radical Expressions

Simplifying radical expressions involves rewriting expressions with radicals in a clearer and simpler form.
The goal is to eliminate any radicals in denominators by using techniques such as multiplying by conjugates. This often leads to expressions with only numerators containing radicals if necessary.
In this process, constants and radicals are simplified as separate terms. After rationalizing the denominator, as seen in this exercise, simplify each part of the fraction individually:

• The constant term: Simplify the division of whole numbers.
• The radical term: Simplify radicals where possible to make calculations cleaner.

This step ensures the simplest possible form of the expression, making it easier for further calculations or interpretations.

Distributive Property

The distributive property is crucial in simplifying expressions, especially when distributing one term over others in a binomial. In the context of multiplying radical expressions, this involves multiplying both terms of a binomial by the rationalizing factor or conjugate.
Let's say we have to multiply \(8(2+\sqrt{6})\):
Apply the distributive property as follows:

• First, distribute 8 to 2: \(8 \times 2 = 16\)
• Next, distribute 8 to \(\sqrt{6}\): \(8 \times \sqrt{6} = 8\sqrt{6}\)

The operation results in \(16 + 8\sqrt{6}\).
By ensuring every term is correctly distributed, radical expressions become accurately arranged, setting the stage for further simplification.