Problem 40 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{4+\sqrt{6}}{\sqrt{2}+\sqrt{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\sqrt{2}+2\sqrt{3}\).

Step 1: Rationalize the Denominator

To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{2}+\sqrt{3}\) would be \(\sqrt{2}-\sqrt{3}\). Then, the expression becomes \(\frac{ (4+\sqrt{6}) (\sqrt{2}-\sqrt{3})}{ (\sqrt{2}+\sqrt{3}) (\sqrt{2}-\sqrt{3})}\).

Step 2: Simplify the Numerator

Expand the numerator using the distributive property. This means first multiplying \(4\) by each quantity inside the parentheses, and then multiplying \(\sqrt{6}\) by each quantity. This results in a numerator of \(4\sqrt{2}-4\sqrt{3}+\sqrt{12}-\sqrt{18}\). The expression \(4\sqrt{2}-4\sqrt{3}+2\sqrt{3}-3\sqrt{2} = \sqrt{2}-2\sqrt{3}\).

Step 3: Simplify the Denominator

The denominator becomes \( (\sqrt{2})^2 - (\sqrt{3})^2 \), which simplifies to \(2 - 3 = -1\).

Step 4: Final Simplification

The division of the expression by \(-1\) leads to the final simplified expression of \(-\sqrt{2}+2\sqrt{3}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra and calculus. When simplifying an expression, our goal is to make it as easy to read and understand as possible. This process often involves combining like terms, removing parentheses, and reducing fractions. For instance, in the problem given, after rationalizing the denominator, the expression was in an expanded form with terms like \(4\sqrt{2}, -4\sqrt{3}\), and others. Combining them further simplified it to \(\sqrt{2} - 2\sqrt{3}\) before final adjustments were made.
A simplified expression is generally easier to work with in further mathematical operations.

• Ensure all like terms are combined.
• Reduce any fractions.
• Consider numerical constants and coefficients in radical expressions.

When dealing with radicals, simplifying expressions can also mean reducing the radical itself if possible, which makes further calculations more straightforward.

Conjugates

A conjugate in mathematics often refers to an expression formed by changing the sign between two terms in a binomial. For example, the conjugate of \(a + b\) is \(a - b\). In the context of rationalizing denominators, the use of conjugates is a practical method to eliminate radicals from the denominator of a fraction.
Using conjugates involves multiplying both the numerator and the denominator by this conjugate expression. This technique leverages the difference of squares formula: \((a+b)(a-b) = a^2-b^2\), which eliminates the radical when the two terms, \(\sqrt{2}+\sqrt{3}\) and its conjugate \(\sqrt{2}-\sqrt{3}\), are multiplied.

• Conjugates are specifically effective for radicals in binomial form.
• They make use of the difference of squares to simplify radical expressions.

Mastering this concept simplifies the algebra involving radicals significantly and is frequently applied in algebra and higher mathematics.

Radicals

Radicals, or root expressions, are common in mathematical problems and include square roots, cube roots, and higher order roots. Simplifying expressions with radicals involves rationalizing them, which means ensuring no radicals remain in the denominator.
Radicals need to be handled carefully as they follow unique arithmetic rules compared to regular numbers.
For instance, \(\sqrt{18}\) can be simplified as \(3\sqrt{2}\) by recognizing that \(18 = 9 \times 2\) and \(\sqrt{9}\) is 3.

• Identifying perfect squares within radical terms aids in simplifying them.
• Combining and simplifying radicals requires attention to equivalent radical expressions.

Understanding how to manipulate these expressions is crucial for success in algebra and calculus, ensuring no radicals in denominators which is important for further calculations and for expressing the final answer in a manageable form.