Problem 46 - Intermediate Algebra

Question

Rationalize each denominator. See Example 4. $$ \frac{\sqrt{3}+\sqrt{4}}{\sqrt{2}-\sqrt{3}} $$

Step-by-Step Solution

Answer

The rationalized expression is \(-\sqrt{6} - 3 - 2\sqrt{2} - 2\sqrt{3}\).

Step 1: Identify the Denominator

The denominator of the provided expression is \( \sqrt{2} - \sqrt{3} \). We need to rationalize this denominator.

Step 2: Multiply by Conjugate

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \( \sqrt{2} - \sqrt{3} \) is \( \sqrt{2} + \sqrt{3} \). Thus, multiply: \[ \frac{(\sqrt{3}+\sqrt{4})(\sqrt{2}+\sqrt{3})}{(\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})} \]

Step 3: Simplify the Denominator

The denominator, \((\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})\), becomes a difference of squares: \[ (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1 \]

Step 4: Expand the Numerator

Expand the numerator \((\sqrt{3}+\sqrt{4})(\sqrt{2}+\sqrt{3})\):\[ \sqrt{3}\sqrt{2} + \sqrt{3}\sqrt{3} + \sqrt{4}\sqrt{2} + \sqrt{4}\sqrt{3} \] Simplify each term: \[ \sqrt{6} + 3 + 2\sqrt{2} + 2\sqrt{3} \]

Step 5: Simplify the Expression

Combine the numerator over the simplified denominator:\[ \frac{\sqrt{6} + 3 + 2\sqrt{2} + 2\sqrt{3}}{-1} \]Simplify the expression by factorizing out \(-1\) from the numerator:\[ -(\sqrt{6} + 3 + 2\sqrt{2} + 2\sqrt{3}) \]

Step 6: Final Expression

The rationalized expression is:\[ -\sqrt{6} - 3 - 2\sqrt{2} - 2\sqrt{3} \]

Key Concepts

Conjugate Multiplication

Conjugate multiplication is a technique used in algebra to simplify expressions, specifically when dealing with radicals. When we want to rationalize a denominator, like in the expression \( \frac{\sqrt{3} + \sqrt{4}}{\sqrt{2} - \sqrt{3}} \), we use the conjugate of the denominator. The conjugate of a binomial \( a - b \) is \( a + b \), and vice versa.

- **Why Use Conjugate Multiplication?** - It helps eliminate radicals in the denominator by transforming them into whole numbers. - By multiplying by the conjugate, you create a difference of squares, which is simpler to work with.
To apply conjugate multiplication, multiply both the numerator and the denominator of the expression by the conjugate of the denominator. This keeps the value of the expression the same since you are effectively multiplying by 1. In our example, we multiply by \( \sqrt{2} + \sqrt{3} \). This process transforms the expression while ensuring the denominator is a rational number.

Difference of Squares

The difference of squares is a powerful algebraic identity that makes simplifying products of conjugates straightforward. It is given by the formula:
\[ a^2 - b^2 = (a + b)(a - b) \]
In our problem, the denominator \((\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3})\) simplifies using the difference of squares:
- When you expand the multiplication: \( (\sqrt{2})^2 - (\sqrt{3})^2 \).- Calculate each square: \( 2 - 3 \).- The result: \( -1 \), a rational number.
The beauty of the difference of squares lies in how efficiently it transforms a more complex multiplication into a single number. It gives a quick path to rationalize denominators when square roots are involved.

Simplifying Radicals

Simplifying radicals is a key skill in algebra that helps make expressions easier to handle. A radical is any expression that includes a square root, cube root, or any higher-order root.

- **Steps for Simplifying Radicals:** 1. Look for perfect squares or perfect cubes within the radical. 2. Rewrite the radical expression as a product of simpler radicals, if possible. 3. Simplify each radical to its simplest form.
In our numerator, \((\sqrt{3} + \sqrt{4})(\sqrt{2} + \sqrt{3})\), we focus on breaking down each term:- \( \sqrt{4} = 2 \), a simple example of simplifying a radical.- Similar steps apply when multiplying and simplifying other mixed radical forms such as \( \sqrt{6} \) and \( 3 \).
Simplifying helps in reducing the complexity of expressions and is crucial when performing operations like adding, subtracting, and rationalizing algebraic fractions.

Algebraic Fractions

Algebraic fractions are fractions that include polynomials in the numerator, the denominator, or both. Rationalizing these fractions often involves strategies like simplifying radicals and using conjugates.

In this exercise, we encounter an algebraic fraction:- The original expression: \( \frac{\sqrt{3} + \sqrt{4}}{\sqrt{2} - \sqrt{3}} \)
- **Steps for Simplification:** - Identify components: numerator = radical sum, denominator = difference of radicals. - Use conjugate multiplication to simplify the denominator. - Perform algebraic operations on the numerator and the denominator.
After simplifying both the numerator and the denominator, the last step is rearranging any changes you've made. In our example, after simplification, the negative sign is factored out to reflect the simplification properly. Effectively simplifying these complex expressions is fundamental for solving algebraic equations and improving mathematical fluency.