Problem 56 - Algebra for College Students

Question

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{-4}{\sqrt{6}-3}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4\sqrt{6}}{3} + 4 \).

Step 1: Identify the Conjugate

The goal is to eliminate the square root from the denominator by multiplying by the conjugate. Since our denominator is \( \sqrt{6} - 3 \), the conjugate is \( \sqrt{6} + 3 \).

Step 2: Multiply Numerator and Denominator

Multiply both the numerator and the denominator by the conjugate. This results in: \[ \frac{-4(\sqrt{6} + 3)}{(\sqrt{6} - 3)(\sqrt{6} + 3)} \]

Step 3: Simplify the Denominator

Use the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\) to simplify the denominator.Substitute \(a = \sqrt{6}\) and \(b = 3\): \[\frac{-4(\sqrt{6} + 3)}{6 - 9} = \frac{-4(\sqrt{6} + 3)}{-3} \]

Step 4: Simplify the Expression

Simplify the fraction by dividing each term in the numerator by \(-3\) separately: \[-4(\sqrt{6}) / -3 + (-4)(3) / -3 = \frac{4\sqrt{6}}{3} + 4 \]

Step 5: Write the Final Simplified Result

Combine the terms to get the final result: \[ \frac{4\sqrt{6}}{3} + 4 \]

Key Concepts

Conjugate in Algebra

When you're trying to rationalize the denominator, understanding the concept of the conjugate is super helpful. A conjugate is simply a pair of numbers or expressions. They have the same terms but opposite signs between them. For example, if you have an expression like \( \sqrt{6} - 3 \), its conjugate would be \( \sqrt{6} + 3 \). The tricky part is replacing the minus with a plus.

So, why do we use the conjugate? That's an easy one!

• To eliminate square roots or radicals from the denominator
• To simplify the expression into a more recognizable form

When you multiply something by its conjugate, you use algebra magic to get rid of those nasty square roots. Always remember multiplying by the conjugate means doing the same operation to both the numerator and the denominator.This is a critical step in making sure that any radicals in the denominator are removed, which is an essential part of simplifying radical expressions.

Difference of Squares

The difference of squares is a nifty algebraic trick that helps us simplify expressions. It follows a simple formula:\[(a - b)(a + b) = a^2 - b^2\]
What this formula means is that if you have two terms multiplied together in the form of a difference of squares, the expression simplifies significantly:

• Identify \( a \) and \( b \) – these are the square root values or numbers you're dealing with.
• Replace \( a \) and \( b \) in the formula.
• Simplify the expression by recognizing that one set of terms cancels out the radical when the squares are taken.

In our example, \( a = \sqrt{6} \) and \( b = 3 \). When we multiply their conjugates, \( (\sqrt{6} - 3) \) and \( (\sqrt{6} + 3) \), the result is perfect due to the difference of squares:\[ \sqrt{6}^2 - 3^2 = 6 - 9 = -3 \]
Using this concept helps turn a radical expression into something much easier to work with, setting you on the path to simplifying the expression fully.

Simplifying Radical Expressions

Simplifying radical expressions is all about getting your expression into the simplest possible form. It's like cleaning up a messy math room! The key is to remember these steps:

• First, rationalize the denominator using the conjugate if there's a square root involved. This will help eliminate the square root from the bottom of the fraction.
• Once the radical is eliminated, focus on simplifying further if possible. This could involve distributing terms or reducing fractions.
• Combine like terms if applicable to make the expression tidier.

In our problem, we worked through eliminating the radical in the denominator using its conjugate, then simplified the resulting expression:\[ \frac{4\sqrt{6}}{3} + 4 \]
This expression is now much cleaner and free of any radicals in the denominator. Practicing these steps will make you a pro at tackling radical expressions, making algebra feel much more approachable!