Problem 53 - College Algebra

Question

Simplify each expression. $$\frac{\sqrt{12 x}}{2+2 \sqrt{3}}$$

Step-by-Step Solution

Answer

\(-\frac{1}{2} \sqrt{3x} + \frac{3}{2} \sqrt{x}\)

Step 1: Factor and Simplify the Square Root

First, factor the expression inside the square root. We know that \(\sqrt{12x} = \sqrt{4 \cdot 3x} = \sqrt{4} \cdot \sqrt{3x} = 2\sqrt{3x}\). So, the expression simplifies to \(\frac{2\sqrt{3x}}{2 + 2\sqrt{3}}\).

Step 2: Simplify the Denominator

Notice that the denominator \(2 + 2\sqrt{3}\) can be simplified by factoring out a 2: \(2(1 + \sqrt{3})\). So the expression becomes \(\frac{2\sqrt{3x}}{2(1 + \sqrt{3})}\).

Step 3: Cancel Common Factors

Since there is a common factor of 2 in the numerator and the denominator, cancel it out.\[\frac{2\sqrt{3x}}{2(1 + \sqrt{3})} = \frac{\sqrt{3x}}{1 + \sqrt{3}}\].

Step 4: Rationalize the Denominator

Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. The conjugate of \(1 + \sqrt{3}\) is \(1 - \sqrt{3}\). So multiply: \(\frac{\sqrt{3x}(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}\).

Step 5: Simplify Using Conjugate Properties

The denominator becomes \((1 + \sqrt{3})(1 - \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2\). The expression is now: \(\frac{\sqrt{3x}(1 - \sqrt{3})}{-2}\).

Step 6: Distribute and Express the Simplified Result

Distribute \(\sqrt{3x}\) in the numerator: \(\frac{\sqrt{3x} - 3\sqrt{x}}{-2}\). Hence, the expression simplifies to: \(-\frac{1}{2} \sqrt{3x} + \frac{3}{2} \sqrt{x}\).

Key Concepts

Rationalizing Denominators

Rationalizing denominators is a technique that helps remove the square root or any radical from the denominator of a fraction. By doing this, the expression is often easier to work with, both conceptually and mathematically.
For example, in our original problem, the denominator is \(1 + \sqrt{3}\). To rationalize it, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 + \sqrt{3}\) is \(1 - \sqrt{3}\).

This multiplication simplifies things. When you multiply \((1 + \sqrt{3})(1 - \sqrt{3})\), you effectively use the difference of squares formula, which results in \(1^2 - (\sqrt{3})^2 = 1 - 3 = -2\). This simplification removes the square root from the denominator.
The rewritten expression, when multiplied by the conjugate, has a much cleaner denominator, making further simplifications and calculations easier to carry out.

Square Roots Simplification

Simplifying square roots involves transforming the expression under the square root into its simplest form. This often involves factoring the number inside the square root to pull out perfect squares.

In the given exercise, the square root \(\sqrt{12x}\) is simplified by factoring 12 as \(4 \cdot 3\), since 4 is a perfect square, we can express \(\sqrt{12x}\) as \(\sqrt{4 \cdot 3x} = \sqrt{4} \cdot \sqrt{3x} = 2\sqrt{3x}\).
By pulling out \(\sqrt{4} = 2\), we simplify \(\sqrt{12x}\) to \(2\sqrt{3x}\). This process not only makes the expression neater but also often reveals factors that can be cancelled out with other elements in the expression, making it simpler overall.

Algebraic Fractions Simplification

In algebraic fractions, simplification usually involves reducing the expression to as simple a form as possible by factoring and canceling common factors between the numerator and the denominator.

Let's take the simplified expression: \(\frac{2\sqrt{3x}}{2(1 + \sqrt{3})}\). Both the numerator and denominator have a factor of 2 that can be cancelled out immediately. Removing this common factor simplifies the expression to \(\frac{\sqrt{3x}}{1 + \sqrt{3}}\).
The key steps in simplifying algebraic fractions include identifying and canceling common factors and then rationalizing the denominator, if necessary, which leads to the expression in its most simplified form.