Problem 46 - College Algebra

Question

In Exercises \(39-48\), rationalize the denominator. $$\frac{5}{\sqrt{3}-1}$$

Step-by-Step Solution

Answer

The rationalized form of the given fraction is \( \frac{5\sqrt{3}}{2} + \frac{5}{2} \).

Step 1: Identify the Conjugate

The conjugate of \( \sqrt{3} - 1 \) is \( \sqrt{3} + 1 \). The conjugate is formed by changing the sign in the middle of the binomial. So, we change the subtraction to addition.

Step 2: Multiply by the Conjugate

Multiply the fraction by the conjugate over itself. This doesn't change the value of the fraction, since any term divided by itself is 1. Our expression becomes: \( \frac{5}{\sqrt{3}-1} * \frac{\sqrt{3}+1}{\sqrt{3}+1} \)

Step 3: Apply Difference of Squares

Apply the difference of squares property, \( (a+b)(a-b) = a^2 - b^2 \), to the denominator. Also distribute the 5 over \( \sqrt{3} + 1 \) in the numerator. We get: \( \frac{ 5\sqrt{3} + 5 }{ (\sqrt{3})^2 - (1)^2 } \)

Step 4: Simplify the Fraction

Simplify the fraction by executing the operations in the denominator and collect the terms. Our simplified fraction is \( \frac{ 5\sqrt{3} + 5 }{ 2 } \). We can write this as two separate fractions: \( \frac{5\sqrt{3}}{2} + \frac{5}{2} \).

Key Concepts

Conjugate

When you encounter a square root in the denominator of a fraction, rationalizing the denominator often involves using the concept of a conjugate. The conjugate is a companion expression that has the same terms as the original but with the opposite sign between them. For example, the conjugate of \( \sqrt{3} - 1 \) is \( \sqrt{3} + 1 \). This switch from a subtraction to an addition, or vice versa, is what defines a conjugate.

Using the conjugate helps remove the square root from the denominator when multiplied. It simplifies expressions and makes them easier to handle. By multiplying the numerator and denominator by the conjugate of the denominator, we essentially "cancel out" the square root through a clever trick you'll learn about next.

Difference of Squares

The difference of squares is a special algebraic formula you can use after identifying a conjugate. It states that \( (a+b)(a-b) = a^2 - b^2 \). This relationship is particularly useful for eliminating square roots from denominators because it converts a more complex expression into a simpler one.

In this specific exercise with the denominator \( \sqrt{3} - 1 \), multiplying by its conjugate \( \sqrt{3} + 1 \) and applying the difference of squares, gives us:

• \( (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 \)
• Calculating this, we have \( 3 - 1 = 2 \)

This transforms the denominator to a simple rational number, without any square roots.
This method not only rationalizes the denominator but also keeps the value of the fraction unchanged, as any number or expression divided by itself is equal to 1.

Simplifying Fractions

Simplifying fractions is all about making fractions as straightforward and compact as possible while keeping their value unchanged. Once you've applied the difference of squares to the denominator and distributed the terms in the numerator, you often end up with a fraction that can be further broken down.

In our example, the step-by-step process left us with the expression \( \frac{5\sqrt{3} + 5}{2} \). To simplify further, split the fraction into two distinct parts:

• \( \frac{5\sqrt{3}}{2} \)
• \( \frac{5}{2} \)

This decomposition can make it easier to interpret and manage the expression, especially when dealing with more complex operations or equations in larger problems.

Remember, it's crucial to simplify fractions whenever possible, as it leads to more elegant solutions and helps in checking the consistency of your work in mathematical problems.