Problem 44 - College Algebra

Question

In Exercises \(39-48\), rationalize the denominator. $$\frac{3}{3+\sqrt{7}}$$

Step-by-Step Solution

Answer

After rationalizing the denominator, we have \(\frac{9-3\sqrt{7}}{2}\).

Step 1: Identify the Conjugate

To begin, identify the conjugate of the denominator. The conjugate of \(3+\sqrt{7}\) is \(3-\sqrt{7}\).

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator of the fraction by the conjugate \((3-\sqrt{7})\). This will give us the following equation: \(\frac{3(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})}\).

Step 3: Simplify the Equation

Next, simplify the equation. The denominator simplifies to \(3^2 - (\sqrt{7})^2\) which equals \(9 - 7 = 2\). The term in the numerator simplifies to \(9-3\sqrt{7}\). Giving us the simplified equation: \(\frac{9-3\sqrt{7}}{2}\).

Key Concepts

Conjugates

When working with radical expressions in the denominator, such as in the fraction \( \frac{3}{3+\sqrt{7}} \), it can be useful to use conjugates. A conjugate is essentially the same binomial expression with the opposite sign between the terms. Here, the conjugate of \( 3 + \sqrt{7} \) is \( 3 - \sqrt{7} \). Conjugates play a key role in rationalizing the denominator because they help eliminate the radical part of the expression.

By multiplying the original expression by its conjugate, specifically both the numerator and denominator, you can simplify the radical component in the denominator into a rational number. This step does not alter the value of the expression because multiplying by the conjugate is the same as multiplying by 1. Understanding conjugates is crucial because they allow for easier manipulation and simplification of radical expressions.

Simplifying Radical Expressions

Radical expressions can often seem complicated, especially when they are present in fractional denominators. Simplifying them is a necessary step to make mathematics clean and tidy. When radicals are part of a denominator, it's standard to rationalize them, making the denominator a rational number.

In the exercise provided, you need to simplify the expression \( \frac{3(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})} \). To accomplish this, multiply the numerator and the denominator by the conjugate to eliminate the radical. In the denominator, the radicals cancel out because of the difference of squares, specifically \( (a+b)(a-b) = a^2 - b^2 \). For the denominator \((3+\sqrt{7})(3-\sqrt{7})\), this results in \( 9 - 7 = 2 \).

The simplified numerator becomes \( 9 - 3\sqrt{7} \). This simplification process makes the expression easier to work with and understand in subsequent mathematical procedures.

Fraction Simplification

Simplifying fractions is an essential skill in mathematics. It involves reducing complex fractions to their simplest forms to make calculations easier and more efficient. In this exercise, the process begins with rationalizing the denominator using the conjugate.

After simplifying the denominator from a complicated expression \((3+\sqrt{7})(3-\sqrt{7})\) down to 2, the entire fraction looks much simpler. The fraction then becomes \( \frac{9-3\sqrt{7}}{2} \).

This final step is crucial because it transforms an unwieldy expression into a clean, manageable one. Fraction simplification ensures the expression is in its most straightforward form, allowing for easy comprehension and further use in mathematical equations or real-world applications. Overall, understanding the simplification process aids students in developing a deeper comprehension of mathematical principles.