Problem 28 - College Algebra

Question

For the following exercises, simplify each expression. $$ \frac{5}{1+\sqrt{3}} $$

Step-by-Step Solution

Answer

\( \frac{-5 + 5\sqrt{3}}{2} \)

Step 1: Identify the Problem

The exercise asks you to simplify the expression \( \frac{5}{1+\sqrt{3}} \). The denominator, \(1 + \sqrt{3}\), contains a square root, so we will need to rationalize it.

Step 2: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of \(1+\sqrt{3}\), which is \(1-\sqrt{3}\). This will eliminate the square root in the denominator.

Step 3: Multiply the Numerator and Denominator

Calculate \( \frac{5}{1+\sqrt{3}} \times \frac{1-\sqrt{3}}{1-\sqrt{3}} = \frac{5(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})} \). Distribute in the numerator: \(5 \times 1 - 5 \times \sqrt{3} = 5 - 5\sqrt{3}\).

Step 4: Simplify the Denominator

Calculate the denominator by using the difference of squares formula: \((1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2\).

Step 5: Final Expression

The simplified expression becomes \( \frac{5 - 5\sqrt{3}}{-2} \). Separate into two fractions: \( \frac{5}{-2} - \frac{5\sqrt{3}}{-2} = -\frac{5}{2} + \frac{5\sqrt{3}}{2} \).

Step 6: Write Final Answer

Combine the two parts to give the final simplified expression: \( \frac{-5 + 5\sqrt{3}}{2} \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves rewriting them in a form that is easier to work with or understand. This process can often involve reducing the complexity of a mathematical phrase, typically by removing radicals from the denominator.

When an expression involves a radical in the denominator, it is often required to rationalize it by eliminating the square root. This makes the expression simpler and more convenient for further calculations.

To simplify \( \frac{5}{1+\sqrt{3}} \), the goal is to remove the square root from the denominator. We do this by multiplying the expression by a special form of 1, which doesn't change the value of the fraction.

Difference of Squares

The difference of squares is a specific algebraic formula used to simplify certain expressions. It states that \((a^2 - b^2) = (a+b)(a-b)\).

This formula is essential when dealing with expressions involving conjugates because it helps to simplify the denominator.

In the exercise, the expression \((1+\sqrt{3})(1-\sqrt{3})\) in the denominator was simplified using this formula:

• \((1)^2 - (\sqrt{3})^2 = 1 - 3\)
• This helps you eliminate the square roots quickly, giving \(-2\) as the result.

Recognizing and applying the difference of squares formula helps rationalize the denominator effectively.

Conjugate in Mathematics

A conjugate in mathematics refers to a pair of binomials such as \((a+b)\) and \((a-b)\). These are identical except for the sign between their terms.

The conjugate is particularly useful when rationalizing the denominator of a fraction because its multiplication can eliminate square roots.

In the given problem, the conjugate of \(1+\sqrt{3}\) is \(1-\sqrt{3}\). By multiplying both the numerator and the denominator by the conjugate, we can simplify the fraction:

• The process involves multiplying \(\frac{5}{1+\sqrt{3}}\) by \(\frac{1-\sqrt{3}}{1-\sqrt{3}}\).
• This gives a new denominator of \((1+\sqrt{3})(1-\sqrt{3}) = -2\), free of square roots.

Using the conjugate is crucial in rationalizing expressions and achieving a simpler form.