Problem 67 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$\frac{1}{\sqrt{3}}-\frac{2}{\sqrt{12}}+2 \sqrt{3}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Second Term

To simplify the term \( \frac{2}{\sqrt{12}} \), first simplify \( \sqrt{12} \). Notice that \( 12 = 4 \times 3 \), so \( \sqrt{12} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \). Therefore, \( \frac{2}{\sqrt{12}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} \).

Step 2: Combine the First and Second Terms

Now we have \( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{3}} \) which simplifies to \( 0 \).

Step 3: Final Simplification

After combining the first two terms to get \( 0 \), we add the final term \( 2\sqrt{3} \). Hence, the expression simplifies to \( 2\sqrt{3} \).

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics, especially when simplifying expressions. They help us find a number which, when multiplied by itself, gives us the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. Recognizing perfect squares, like 4, 9, 16, is helpful in simplifying square root expressions.
To simplify square roots effectively:

• Break down the number under the square root into its prime factors.
• Identify and pull out pairs of numbers since they equate to a whole number when squared.
• Multiply these whole numbers outside the square root sign while leaving any unpaired numbers inside.

This approach was demonstrated in the original exercise where the square root of 12 was simplified to \( 2\sqrt{3} \) by recognizing that 12 is the product of 4 and 3, with 4 being a perfect square.

The Role of Rationalization

Rationalization is the process of eliminating irrational numbers, such as square roots, from the denominator of a fraction. By transforming the denominator into a rational number, mathematical operations become simpler. This is essential for simplifying expressions as it provides a cleaner, more accessible form.
To rationalize an expression:

• Multiply both the numerator and the denominator by the conjugate or its simplest form leading to a rational denominator.
• For simple roots, multiplying by the square root itself is often enough to eliminate the root from the denominator.
• Simplify the resulting fraction.

In the exercise, rationalization led to equalizing terms like \( \frac{1}{\sqrt{3}} \) and \( \frac{2}{\sqrt{12}} \) by converting them both to \( \frac{1}{\sqrt{3}} \), making it easy to combine and simplify further.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms, canceling terms, and/or calculating numerical results to make the expression as simple as possible. Mastering this skill is key for solving complex algebraic problems efficiently.
The process includes:

• Identifying and grouping similar terms or roots.
• Performing mathematical operations like addition or subtraction among like terms.
• Using factorization to further simplify or make patterns evident.

In the step-by-step solution provided in the exercise, simplifying the term \( \frac{2}{\sqrt{12}} \) to \( \frac{1}{\sqrt{3}} \) allowed for cancellation of terms. Eventually, this process reduced the expression to a single term, \( 2\sqrt{3} \), demonstrating the effectiveness of simplification techniques.