Problem 28 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{5}{2} + \frac{5\sqrt{3}}{2}\).

Step 1: Identify the Need for Rationalization

The denominator of the expression \(\frac{5}{1+\sqrt{3}}\) is not rational. We need to multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.

Step 2: Multiply by the Conjugate

Identify the conjugate of \(1+\sqrt{3}\), which is \(1-\sqrt{3}\). Multiply both the numerator and the denominator by this conjugate:\[ \frac{5}{1+\sqrt{3}} \times \frac{1-\sqrt{3}}{1-\sqrt{3}} = \frac{5(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})} \]

Step 3: Simplify the Denominator

Simplify the expression \((1+\sqrt{3})(1-\sqrt{3})\) using the difference of squares formula, \((a+b)(a-b) = a^2 - b^2\):\[ (1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \]

Step 4: Simplify the Numerator

Expand the numerator of the expression:\[ 5(1-\sqrt{3}) = 5 \times 1 - 5 \times \sqrt{3} = 5 - 5\sqrt{3} \]

Step 5: Combine and Simplify the Expression

Combine the simplified numerator and denominator:\[ \frac{5-5\sqrt{3}}{-2} \]Distribute the negative sign in the denominator across the numerator:\[ -\frac{5}{2} + \frac{5\sqrt{3}}{2} \]So, the simplified expression is:\(-\frac{5}{2} + \frac{5\sqrt{3}}{2}\).

Key Concepts

Conjugate

When you come across an expression in the denominator that contains a square root, like in \( \frac{5}{1+\sqrt{3}} \), simplifying can be tricky unless you use the concept of the **conjugate**.

A conjugate involves swapping the sign between two terms. For a term like \( 1+\sqrt{3} \), its conjugate is \( 1-\sqrt{3} \). Multiplying by the conjugate helps to eliminate the square root from the denominator.

Here’s why it works:

• Multiplying a term by its conjugate results in the difference of squares (more on that in the next section).
• This multiplication leads to a whole number, which rationalizes, or "makes rational," the denominator.
• Multiplying the expression by \( \frac{1-\sqrt{3}}{1-\sqrt{3}} \) is like multiplying by one, so it doesn’t change the value of the expression.

Looking at the problem, you’ll see that using the conjugate helps clear out the irrational part of the denominator, making further steps in simplification possible and easier.

Difference of Squares

The difference of squares is a powerful algebraic identity that can simplify expressions involving squares.

The identity is written as follows: \((a+b)(a-b) = a^2 - b^2\). This identity helps you deal with expressions like \((1+\sqrt{3})(1-\sqrt{3})\). Using this formula, we rearrange and simplify:

• First, recognize \( a = 1 \) and \( b = \sqrt{3} \).
• Apply the identity: \( 1^2 - (\sqrt{3})^2 = 1 - 3 \).
• The result is \(-2\), a simple whole number.

By using the difference of squares, you effectively turn a complex expression into a simpler, rational expression.
Clearer calculations generally follow, and you're left with a denominator that's much easier to work with in subsequent steps.

Simplification

Simplification in algebra means reducing an expression to its simplest form. After using the conjugate and the difference of squares, you reduce complexity, especially if that expression looks intimidating because of radicals or roots.

Once you've made the denominator rational by applying the difference of squares, focus on simplifying the entire fraction. Using our example \( \frac{5 - 5\sqrt{3}}{-2} \):

• Distribute the terms in the numerator across the simplified whole number \(-2\).
• This gives you \(-\frac{5}{2} + \frac{5\sqrt{3}}{2} \).

Remember:

• Although the fraction looks a bit different than the original, these represent the same value. We're just simplifying for ease and clarity.
• Pay attention to signs during simplification to keep the proper mathematical description.

Simplification is about ensuring that the expression is not only correct but also easier to use in solving additional problems or calculations.