Problem 32 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{3}{\sqrt{7}-4}\)

Step-by-Step Solution

Answer

The rationalized form is \(-\frac{\sqrt{7}}{3} - \frac{4}{3}\).

Step 1: Identify the Conjugate

To rationalize the denominator of the expression \( \frac{3}{\sqrt{7}-4} \), we first need to identify the conjugate of the denominator. The denominator is \( \sqrt{7} - 4 \), hence its conjugate is \( \sqrt{7} + 4 \).

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate from Step 1. This gives us:\[\frac{3}{\sqrt{7} - 4} \times \frac{\sqrt{7} + 4}{\sqrt{7} + 4} = \frac{3(\sqrt{7} + 4)}{(\sqrt{7} - 4)(\sqrt{7} + 4)}\]

Step 3: Simplify the Denominator

The denominator is a difference of squares, \( (a - b)(a + b) = a^2 - b^2 \). So, simplify it:\[(\sqrt{7})^2 - 4^2 = 7 - 16 = -9 \]

Step 4: Simplify the Numerator

Expand the numerator:\[3(\sqrt{7} + 4) = 3\sqrt{7} + 12\]

Step 5: Combine and Simplify the Expression

Combine the simplified numerator and denominator:\[\frac{3\sqrt{7} + 12}{-9}\]Divide each term by \(-9\) to simplify the fraction:\[-\frac{3\sqrt{7}}{9} - \frac{12}{9} = -\frac{\sqrt{7}}{3} - \frac{4}{3} \]

Step 6: Final Simplified Expression

The rationalized form of \( \frac{3}{\sqrt{7} - 4} \) is:\[-\frac{\sqrt{7}}{3} - \frac{4}{3} \]

Key Concepts

Conjugates

In mathematics, the concept of conjugates is an essential tool, especially when rationalizing denominators involving square roots. The term conjugate refers to two expressions that are identical, except their middle operation sign is flipped, usually between a plus and minus. For example, if you have

• an expression like \( a + b \), the conjugate would be \( a - b \).
• Similarly, for \( \sqrt{7} - 4 \), the conjugate is \( \sqrt{7} + 4 \).

When you multiply a number by its conjugate, it removes any radicals from the denominator due to a technique called the "difference of squares". This simplifies calculations significantly. It's important to remember that conjugates are primarily used for expressions that have a square root or other term with a variable when combined using subtraction or addition. By multiplying by the conjugate, we essentially make it possible to rationalize the denominator, ensuring that no radicals remain.

Difference of Squares

The difference of squares is a powerful algebraic identity that states

• \((a - b)(a + b) = a^2 - b^2\).

This identity is incredibly useful when simplifying expressions involving conjugates. For example, consider the denominator from our expression:

• \((\sqrt{7} - 4)(\sqrt{7} + 4)\). When applied here, \(a\) is \(\sqrt{7}\) and \(b\) is 4.

Applying the identity, we calculate:

• \(a^2 - b^2 = (\sqrt{7})^2 - 4^2 = 7 - 16 = -9\).

The result is not only simplified, but the radicals are removed, making further arithmetic processing easier. Recognizing when to use the difference of squares offers a strong advantage in manipulating and simplifying complex algebraic fractions.

Simplifying Radicals

Simplifying radicals means making them as simple as possible by removing any square roots from the denominator. The goal is to prepare the expression in an easily understandable and convenient form. Let's break it down more:

• To simplify a radical, you often multiply by a form of 1 that involves the conjugate. This gets rid of any square roots in the denominator.
• After multiplying, simplify both the numerator and denominator as much as possible.

After rationalizing and multiplying by the conjugate, you might have a fraction like \(\frac{3\sqrt{7} + 12}{-9}\). From here, we break it down to

• \(-\frac{3\sqrt{7}}{9} - \frac{12}{9}\).

By dividing each term by the common factor, division simplifies the expression further:

• \(-\frac{\sqrt{7}}{3} - \frac{4}{3}\).

This final expression is clear and in a simpler form, showing no radicals in the denominator, which is crucial in many mathematical applications, including various levels of calculus and algebra.