Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\frac{\sqrt{12 x}}{2+2 \sqrt{3}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3\sqrt{x} - \sqrt{3x}}{2} \).

Step 1: Identify the problem

We are given the expression \( \frac{\sqrt{12x}}{2+2 \sqrt{3}} \) and need to simplify it. The first step is to look for a pattern where we can rationalize the denominator to remove the square root from it.

Step 2: Simplify the expression inside the square root

First, we'll simplify \( \sqrt{12x} \). Notice that 12 can be factored as 4 and 3, that is \( \sqrt{12x} = \sqrt{4 \cdot 3 \cdot x} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{x} = 2\sqrt{3x} \). This simplifies the expression to \( \frac{2\sqrt{3x}}{2+2\sqrt{3}} \).

Step 3: Simplify the denominator

Factor out a 2 from the denominator: \( 2 + 2 \sqrt{3} = 2(1 + \sqrt{3}) \). Now the expression looks like \( \frac{2\sqrt{3x}}{2(1+\sqrt{3})} \).

Step 4: Cancel common factors

Since there is a 2 in both the numerator and denominator, we can cancel them: \( \frac{2\sqrt{3x}}{2(1+\sqrt{3})} = \frac{\sqrt{3x}}{1+\sqrt{3}} \).

Step 5: Rationalize the denominator

Multiply both the numerator and denominator by the conjugate of the denominator, \( 1-\sqrt{3} \), to remove the square root. \[ \frac{\sqrt{3x}}{1+\sqrt{3}} \cdot \frac{1-\sqrt{3}}{1-\sqrt{3}} = \frac{\sqrt{3x}(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})} = \frac{\sqrt{3x}(1-\sqrt{3})}{1^2-(\sqrt{3})^2} = \frac{\sqrt{3x}(1-\sqrt{3})}{1-3} = \frac{\sqrt{3x}(1-\sqrt{3})}{-2} \]

Step 6: Simplify the final expression

Now distribute \( \sqrt{3x} \) in the numerator: \( \frac{\sqrt{3x}\cdot1 - \sqrt{3x}\cdot\sqrt{3}}{-2} = \frac{\sqrt{3x} - 3\sqrt{x}}{-2} \).This simplifies to: \[ -\frac{\sqrt{3x} - 3\sqrt{x}}{2} = \frac{3\sqrt{x} - \sqrt{3x}}{2} \].

Step 7: Conclusion

The expression \( \frac{\sqrt{12 x}}{2+2 \sqrt{3}} \) simplifies to \( \frac{3\sqrt{x} - \sqrt{3x}}{2} \).

Key Concepts

Rationalizing Denominators

Rationalizing the denominator is an important technique in simplifying expressions with square roots. The goal is to make sure there are no square roots in the denominator of a fraction because it's a standard in math to have a simpler and more acceptable form. To achieve this with a denominator like \(1 + \sqrt{3}\), we multiply both the numerator and the denominator by the conjugate of the denominator.

• The conjugate of \(1 + \sqrt{3}\) is \(1 - \sqrt{3}\).
• Multiplying the original fraction \( \frac{\sqrt{3x}}{1+\sqrt{3}} \) by \( \frac{1 - \sqrt{3}}{1 - \sqrt{3}} \) eliminates the square root in the denominator.
• This works because \( (1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \).

Once the square root is eliminated from the denominator, the expression looks neat and simplifies further algebraic operations.

Square Roots

Square roots are extracted by finding a number that, when multiplied by itself, gives the original number. Simplifying square roots can make expressions easier to handle. For instance, in the problem we have \(\sqrt{12x}\). By breaking down 12 into its prime factors, we recognize:

• 12 can be expressed as \(4 \times 3\).
• This allows \(\sqrt{12x} = \sqrt{4 \cdot 3 \cdot x} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{x}\), which simplifies to \(2\sqrt{3x}\).

Simplifying square roots is really about identifying perfect squares and using properties of roots to split the expression in simple terms. It simplifies calculations and further operations in algebra.
Breaking down complex square root expressions helps in easier manipulation and further simplifications.

Algebraic Expressions

An algebraic expression consists of constants, variables, and the operations performed on them. Understanding how to manipulate these expressions is key to solving algebra problems. In our problem, we dealt with expressions like \(\frac{2\sqrt{3x}}{2(1+\sqrt{3})}\).

• Simplifying involves canceling common factors. For instance, the 2 in the numerator and denominator can be canceled.
• We then end up with \(\frac{\sqrt{3x}}{1+\sqrt{3}}\), showcasing simplified form using basic algebraic manipulations.
• The process also involves multiplying by the conjugate, a common technique when dealing with radicals, to rationalize the expression.

Understanding these manipulations is crucial for confidently solving algebra problems and achieving simplified, easy-to-understand results. This knowledge streamlines complex problems, making them more approachable and solvable.