Problem 65 - Algebra for College Students

Question

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{6}}{3 \sqrt{2}+2 \sqrt{3}}\)

Step-by-Step Solution

Answer

\(\sqrt{3} - \sqrt{2}\)

Step 1: Identify Rationalization Strategy

To rationalize the denominator, we need to eliminate the square roots from the denominator. In this case, since the denominator is of the form \(a\sqrt{b} + c\sqrt{d}\), we'll multiply both the numerator and the denominator by its conjugate, \(a\sqrt{b} - c\sqrt{d}\). Here, the conjugate of \(3\sqrt{2} + 2\sqrt{3}\) is \(3\sqrt{2} - 2\sqrt{3}\).

Step 2: Multiply by Conjugate

Multiply both the numerator and the denominator by the conjugate \(3\sqrt{2} - 2\sqrt{3}\): \[\frac{\sqrt{6}(3\sqrt{2} - 2\sqrt{3})}{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})}\]

Step 3: Simplify the Numerator

Distribute \(\sqrt{6}\) in the numerator: \[\sqrt{6}(3\sqrt{2}) - \sqrt{6}(2\sqrt{3})\]Which simplifies to: \[3\sqrt{12} - 2\sqrt{18}\]Continue simplifying the square roots: \(\sqrt{12} = 2\sqrt{3}\) and \(\sqrt{18} = 3\sqrt{2}\), so: \[3(2\sqrt{3}) - 2(3\sqrt{2}) = 6\sqrt{3} - 6\sqrt{2}\]

Step 4: Expand the Denominator

Use the difference of squares formula \((x+y)(x-y) = x^2 - y^2\):\[(3\sqrt{2})^2 - (2\sqrt{3})^2 = 9 \times 2 - 4 \times 3\]This simplifies to: \[18 - 12 = 6\]

Step 5: Combine and Simplify the Expression

Now combine the results:\[\frac{6\sqrt{3} - 6\sqrt{2}}{6}\]Divide each term in the numerator by the denominator:\[\frac{6\sqrt{3}}{6} - \frac{6\sqrt{2}}{6} = \sqrt{3} - \sqrt{2}\]

Step 6: Write the Final Answer

The expression is now simplified, and the denominator is rationalized. The simplified form is \(\sqrt{3} - \sqrt{2}\).

Key Concepts

Conjugate Multiplication

When dealing with expressions that contain square roots, particularly in the denominator, conjugate multiplication becomes a powerful tool. The goal of rationalizing the denominator is to remove any radicals (square roots) so the expression is easier to work with, especially for further algebraic manipulation.
The conjugate of a binomial expression that involves square roots is simply the same expression with the opposite sign between the terms. For example, if you have a denominator like \(3\sqrt{2} + 2\sqrt{3}\), its conjugate will be \(3\sqrt{2} - 2\sqrt{3}\).

• Multiplying by the conjugate often turns the denominator into a simpler, rational number.
• This process involves using the fact that \((x+y)(x-y) = x^2 - y^2\), eliminating the square roots in the denominator.

Conjugate multiplication makes equations cleaner and easier to solve. It changes complex fractions into simpler, more manageable forms.

Difference of Squares

The difference of squares is a specific pattern used in algebra to simplify expressions, and it is instrumental when rationalizing denominators involving radicals.
This technique is useful because of its ability to transform a product of sums and differences into a simpler expression: \[(a+b)(a-b) = a^2 - b^2\].
In the context of the original exercise, when using the conjugate on the denominator \((3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})\), the difference of squares formula simplifies the denominator to:\[ (3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6.\]

• The square root terms cancel each other out, leaving behind simple arithmetic.
• This results in a rationalized denominator, which simplifies further calculations.

Understanding the difference of squares helps recognize when an expression can be streamlined, making algebraic fractions much easier to work with.

Simplifying Radicals

Simplifying radicals is essential when working with expressions containing square roots, as it breaks down more complex roots into easily manageable forms. The goal is to express each radical in its simplest form by factoring out squares.

• A radical like \(\sqrt{12}\) simplifies to \(2\sqrt{3}\) because 12 can be broken down into its prime factors, resulting in \(2^2 \times 3\).
• Similarly, \(\sqrt{18}\) simplifies to \(3\sqrt{2}\).

In the numerator of our original expression, applying these simplifications results in the transformation:\[3\sqrt{12} - 2\sqrt{18} = 6\sqrt{3} - 6\sqrt{2}.\]
By rewriting radicals in such a simple form, you make the expression more straightforward to handle or further simplify. This process reduces potential errors and makes it easier to combine like terms or complete further algebraic operations.