Problem 102 - Algebra 2

Question

Simplify. Rationalize all denominators. $$ \frac{-2+\sqrt{8}}{-3-\sqrt{2}} $$

Step-by-Step Solution

Answer

The simplified form of the rational expression is \(\frac{2 -3\sqrt{8} + 2\sqrt{2}}{7}\).

Step 1: Identify the Conjugate

The conjugate of the denominator \(-3-\sqrt{2}\) is \(-3+\sqrt{2}\). We obtain this by simply changing the sign in the middle.

Step 2: Multiply by the Conjugate

Multiply the numerator and denominator by the conjugate \(-3+\sqrt{2}\) to rationalize the denominator. So, \[\frac{-2+\sqrt{8}}{-3-\sqrt{2}} \times \frac{-3+\sqrt{2}}{-3+\sqrt{2}}\]

Step 3: Simplify the Expression

Now, use the rule \((a - b)(a + b) = a^2 - b^2\) to simplify the expression, \[\frac{2(-3+\sqrt{2}) + (\sqrt{8})(-3+\sqrt{2})}{(-3)^2 - (\sqrt{2})^2}\]Further steps of simplification will result to: \[\frac{-6 + 2\sqrt{2} - 3\sqrt{8} + 2\sqrt{16}}{9 - 2} = \frac{-6 + 2\sqrt{2} - 3\sqrt{8} + 8}{7} \]By simplifying, we obtain the result \(\frac{2 -3\sqrt{8} + 2\sqrt{2}}{7}\).

Key Concepts

Conjugate Multiplication

When dealing with radicals in the denominator, we often use a technique called conjugate multiplication. The goal is to eliminate the radicals so the denominator becomes a rational number. The conjugate of a binomial expression \( a - b \) or \( a + b \) is formed by changing the sign between the two terms. So for the denominator \(-3-\sqrt{2}\), its conjugate will be \(-3+\sqrt{2}\).

We multiply both the numerator and denominator of the fraction by this conjugate. This means, the expression \(\frac{-2+\sqrt{8}}{-3-\sqrt{2}}\) becomes:

• Numerator: \((-2+\sqrt{8})(-3+\sqrt{2})\)
• Denominator: \((-3-\sqrt{2})(-3+\sqrt{2})\)

The effect of the conjugate in the denominator is to transform it into a difference of squares \((a - b)(a + b) = a^2 - b^2\). This operation removes the square root, making the denominator a simple integer.

Simplifying Radicals

After applying conjugate multiplication, you often end up with expressions that have radicals. To make further simplifications easier, it's important to simplify these radicals where possible.

Take the radical \( \sqrt{8} \) for example. It can be simplified by finding the largest perfect square factor.

• \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \)

This simplification helps to break down calculations into smaller parts, making multiplication and addition or subtraction easier.

The process of simplifying radicals is about expressing them in the simplest or most simplified form possible, allowing us to easily combine like terms and simplify expressions overall.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. They function similarly to normal fractions but can involve more complex operations due to variables and radicals.

To simplify algebraic fractions, look for opportunities to simplify both the numerator and the denominator separately. Once you've simplified radicals and used conjugate multiplication, you might find common terms that can be reduced, or simplified further, to ensure the expression is in its simplest form possible.

In the exercise provided, after performing all these actions, you reach a point where the expression looks like \( \frac{2 - 3\sqrt{8} + 2\sqrt{2}}{7} \). Checking if the coefficients and constants in both the numerator and the denominator can be simplified will help ensure you've simplified the fraction properly.