Problem 65 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{2-\sqrt{11}}{6}\)

Step-by-Step Solution

Answer

Rationalized form: \( \frac{-7}{6(2 + \sqrt{11})} \).

Step 1: Understand the Problem

We need to rationalize the numerator of the given fraction, \( \frac{2 - \sqrt{11}}{6} \). This means we have to eliminate the square root from the numerator.

Step 2: Identify the Conjugate

To rationalize the numerator, multiply the numerator and the denominator by the conjugate of the numerator, which is \( 2 + \sqrt{11} \).

Step 3: Multiply the Numerator by its Conjugate

Multiply \( (2 - \sqrt{11})(2 + \sqrt{11}) \) which uses the formula \((a-b)(a+b) = a^2 - b^2\).

Step 4: Calculate the Result of Conjugate Multiplication

Applying \( a = 2 \) and \( b = \sqrt{11} \), we get \( 2^2 - (\sqrt{11})^2 = 4 - 11 = -7 \).

Step 5: Multiply the Denominator

Multiply the denominator by the conjugate as well. The denominator becomes \( 6(2 + \sqrt{11}) \).

Step 6: Write the Rationalized Expression

The rationalized form of the expression is \( \frac{-7}{6(2 + \sqrt{11})} \).

Key Concepts

Conjugate Multiplication

When you have a square root in a fraction and you want to rationalize the numerator, conjugate multiplication can be your best friend. The concept is straightforward: if a fraction has a numerator of the form \(a - \sqrt{b}\), then the conjugate is \(a + \sqrt{b}\). By multiplying both the numerator and the denominator by this conjugate, you're applying the difference of squares formula \((a-b)(a+b) = a^2 - b^2\). This neat trick helps eliminate the square root in the numerator.

• Example: For the fraction \(\frac{2-\sqrt{11}}{6}\)
• Conjugate of the numerator: \(2 + \sqrt{11}\)
• Multiply both the numerator and denominator by \(2 + \sqrt{11}\)

This multiplication leads to a simplified number where the problematic square root is gone from the numerator, making it easier to work with.

Square Roots

Square roots can sometimes complicate mathematical expressions, especially when they appear in numerators. In our exercise, the numerator \(2 - \sqrt{11}\) contains a square root. To rationalize it, we need to eliminate this square root.

• Square roots appear as expressions under a radical sign.
• They can often make fractions look more complex than they are.
• Removing square roots from numerators and denominators simplifies calculations and comparisons.

Understanding how to manipulate square roots using methods like multiplication by conjugates is crucial in algebra, ensuring expressions can be rewritten in more manageable forms.

Fraction Simplification

Simplifying fractions to their most basic form not only makes them easier to read but also easier to work with in subsequent calculations. In our example, once we rationalized the numerator using conjugation, we arrived at the fraction \(\frac{-7}{6(2 + \sqrt{11})}\). Here's what to consider:

• Accuracy: Ensure that all square roots are properly addressed in the simplification process.
• Multiplication: Make sure to multiply both parts by the conjugate to maintain equality in the expression.
• Simplicity: The simplified fraction should be more intuitive and devoid of square roots in the numerator for easier implementation in solving other problems.

Fraction simplification through techniques such as rationalizing numerators enhances clarity and precision in mathematical solutions.