Problem 49 - Algebra and Trigonometry

Question

Rationalize the denominator. $$ \frac{13}{3+\sqrt{11}} $$

Step-by-Step Solution

Answer

Hence, the rationalized form of the given expression is \(-(39 - 13\sqrt{11})\).

Step 1: Identify the Conjugate

The conjugate of the denominator \((3+\sqrt{11})\) is \((3-\sqrt{11})\).

Step 2: Multiply by Conjugate

Multiply both the numerator and the denominator by the conjugate \((3-\sqrt{11})\). The expression becomes: \[\frac{13 \cdot (3-\sqrt{11})}{(3+\sqrt{11}) \cdot (3-\sqrt{11})}\]

Step 3: Simplify Result

Now we simplify the numerator and denominator. The numerator simplifies to \(39 - 13\sqrt{11}\). The denominator simplifies to \(3^2 - (\sqrt{11})^2\) because \((a-b)\cdot (a+b) = a^2 - b^2\). Therefore, the simplified form is: \[\frac{39 - 13\sqrt{11}}{9 - 11} = -(39 - 13\sqrt{11})\]

Key Concepts

Conjugate in Algebra

In algebra, the conjugate of a binomial expression like \(3+\sqrt{11}\) is the expression \(3-\sqrt{11}\). Conjugates are used in various mathematical processes, especially when dealing with radical expressions. The beauty of conjugates is that when you multiply a binomial by its conjugate, you utilize the difference of squares formula:

• \((a+b)(a-b) = a^2 - b^2\)

This multiplication results in rational numbers when dealing with irrational components, because it eliminates the square roots.
For the expression \(\frac{13}{3+\sqrt{11}}\), the conjugate \(3-\sqrt{11}\) helps to rationalize the denominator. By multiplying both the numerator and denominator by this conjugate, you transform the denominator into a rational number. This step simplifies calculations and is crucial in mathematical operations where radicals in denominators are undesirable.

Algebraic Simplification

Algebraic simplification plays a key role in transforming complex expressions into more manageable forms. When multiplying the numerator and denominator of \(\frac{13}{3+\sqrt{11}}\) by the conjugate \(3-\sqrt{11}\), an important simplification occurs.

• For the numerator: \((13)(3-\sqrt{11}) = 39 - 13\sqrt{11}\)
• For the denominator, using the difference of squares: \((3+\sqrt{11})(3-\sqrt{11}) = 9 - 11\)

The denominator becomes \(-2\), a rational number. Simplification transforms the expression to \(\frac{39 - 13\sqrt{11}}{-2}\). This algebraic transformation not only rationalizes but also neatly organizes the expression into a simpler, more conventional algebraic form that's easier to work with.

Radical Expressions

Radical expressions include terms with roots, such as \(\sqrt{11}\), which is an irrational number. The challenge with radicals is that they often result in complex calculations, especially when present in denominators.
In expressions where radicals are in the denominator, it is common to "rationalize" them. This involves transforming the expression so that the denominator no longer contains radicals.
By multiplying by the conjugate, as shown in the exercise \(\frac{13}{3+\sqrt{11}}\), you effectively eliminate the radical from the denominator. This process not only simplifies the calculation but also adheres to mathematical convention. Rationalizing denominators helps present the expression in a form that's simpler to interpret and solve, especially when further mathematical operations are needed. Understanding how to handle radical expressions is crucial for building a strong algebraic foundation. It allows you to manipulate and streamline expressions for easier handling in algebraic equations.