Problem 40 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{-3}{\sqrt{6}-2}\)

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{-3\sqrt{6}}{2} - 3 \).

Step 1: Identify the problem

We need to rationalize the denominator of the fraction \( \frac{-3}{\sqrt{6}-2} \). This means we need to eliminate the square root from the denominator.

Step 2: Recalling the conjugate method

To rationalize a binomial denominator like \( \sqrt{6} - 2 \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{6} + 2 \). The expression remains equivalent because we are essentially multiplying by 1.

Step 3: Multiply the numerator and the denominator by the conjugate

Multiply both the numerator and the denominator of \( \frac{-3}{\sqrt{6}-2} \) by \( \sqrt{6} + 2 \) to get: \[ \frac{-3(\sqrt{6} + 2)}{(\sqrt{6} - 2)(\sqrt{6} + 2)} \].

Step 4: Simplify the denominator

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

Step 5: Simplify the numerator

Distribute \(-3\) across \( \sqrt{6} + 2 \):\[ -3(\sqrt{6}) + (-3)(2) = -3\sqrt{6} - 6 \].

Step 6: Final simplified fraction

Combine the simplified numerator and denominator to form:\[ \frac{-3\sqrt{6} - 6}{2} \].

Step 7: Distribute the division

Distribute the division across the terms in the numerator:\[ \frac{-3\sqrt{6}}{2} - \frac{6}{2} \].

Step 8: Simplify each term

Simplify \(-\frac{6}{2}\) to \(-3\) to obtain the final rationalized expression:\[ \frac{-3\sqrt{6}}{2} - 3 \].

Key Concepts

Conjugate Method

The conjugate method is a powerful tool used in mathematics to simplify expressions, particularly when dealing with binomial expressions that include irrational numbers, such as square roots. In the context of rationalizing denominators, the conjugate of a binomial expression is essentially just the same expression but with the opposite sign between the two terms.
This method shines when we have a term like \( \sqrt{6} - 2 \) in the denominator. The conjugate of \( \sqrt{6} - 2 \) is \( \sqrt{6} + 2 \). By multiplying the numerator and the denominator by this conjugate, the expression remains mathematically equivalent; this process exploits a foundational algebraic identity.
Multiplying by the conjugate simplifies the square root in the denominator, allowing us to achieve a rational number. This technique is akin to using a mathematical loophole to sidestep the complexity of rooted denominators, easing the path to simplification.

Difference of Squares

The difference of squares is a vital algebraic concept used extensively to simplify binomials where there are squares involved. It's a formula that looks like \((a-b)(a+b) = a^2 - b^2\).
In this rationalization problem, when we multiply the expression \((\sqrt{6} - 2)(\sqrt{6} + 2)\), we leverage the difference of squares formula. The resulting expression \((\sqrt{6})^2 - (2)^2\) simplifies to \(6 - 4\).
The ease and elegance of this concept lie in its ability to strip away the square roots or any form of radicals, resulting in a simple, easily manageable integer or rational number. It's a nifty technique that underscores the importance of recognizing patterns and relationships within algebraic expressions.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest or most efficient form. In arithmetic and algebra, this means performing operations to minimize the complexity of the expression. Following the conjugate method, simplification continues to ensure the final expression is as neat as possible.
After dealing with the denominator, attention turns to the numerator. Multiplying \(-3\) by \(\sqrt{6} + 2\) involves distributing or multiplying each term separately, which results in \(-3\sqrt{6} - 6\).
Further simplification occurs when the resulting terms in the expression are broken down – dividing and combining terms wherever possible to ensure the cleanest outcome. The final simplified expression \(\frac{-3\sqrt{6}}{2} - 3\) exemplifies this process thoroughly and accurately. Simplification, therefore, is not merely a task but a discipline that ensures the precision and efficiency of mathematical solutions.