Problem 36 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}}\)

Step-by-Step Solution

Answer

Rationalized expression is \(-\sqrt{6} - 2\sqrt{2} + 2\sqrt{3} + 3\).

Step 1: Understand Rationalization

Rationalization of a denominator involves removing any square roots or irrational numbers from the denominator. This is accomplished by multiplying the numerator and denominator by the conjugate of the denominator.

Step 2: Identify the Conjugate

The conjugate of the denominator \( \sqrt{2} + \sqrt{3} \) is \( \sqrt{2} - \sqrt{3} \). Conjugates have the same components but opposite signs between them.

Step 3: Multiply by the Conjugate

Multiply both the numerator and the denominator by \( \sqrt{2} - \sqrt{3} \):\[\frac{(\sqrt{3} + \sqrt{4})(\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})}\]

Step 4: Simplify the Denominator

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

Step 5: Expand and Simplify the Numerator

Expand the numerator using distributive property:\[(\sqrt{3} + \sqrt{4})(\sqrt{2} - \sqrt{3}) = \sqrt{3}\sqrt{2} - \sqrt{3}\sqrt{3} + \sqrt{4}\sqrt{2} - \sqrt{4}\sqrt{3}\]which simplifies to:\[\sqrt{6} - 3 + 2\sqrt{2} - 2\sqrt{3}\]Simplify to get: \[\sqrt{6} + 2\sqrt{2} - 2\sqrt{3} - 3\]

Step 6: Final Simplification

Divide each term in the numerator by the -1 in the denominator:\[-(\sqrt{6} + 2\sqrt{2} - 2\sqrt{3} - 3) = -\sqrt{6} - 2\sqrt{2} + 2\sqrt{3} + 3\]

Key Concepts

Conjugate

Rationalizing a denominator often requires finding the conjugate. The conjugate is a simple yet powerful tool. It involves changing the sign between two terms in a binomial expression.
For example, if you have a term like \( \sqrt{2} + \sqrt{3} \), the conjugate would be \( \sqrt{2} - \sqrt{3} \). By multiplying by the conjugate, you can eliminate square roots in the denominator.
The process involves:

• Multiplying the numerator and denominator by this conjugate.
• The denominator, initially with roots, transforms into an expression without roots.

Understanding conjugates is essential for simplifying expressions and finding exact values efficiently. Every time you see a term with a square root in the denominator, remember to consider the conjugate—it’s your key to simplification.

Difference of Squares

The difference of squares is a fundamental algebraic identity, very useful when dealing with conjugates.
This concept states that the product of a sum and difference of the same two terms results in a expression with no square roots:
\[(a + b)(a - b) = a^2 - b^2\]

• When applied, the square roots in the expression vanish because each term squares out the roots.
• This technique greatly simplifies expressions, especially in terms of rationalizing denominators.

Using the difference of squares not only helps in simplifying but also user in handling quadratic equations. Every time you face a sum or difference with square roots, remember this concept—it's a shortcut to clarity.

Distributive Property

The distributive property is a very versatile tool in algebra. It's often used to simplify expressions, including those in fractions where one needs to apply the property to multiply through terms with roots.
The distributive property states that:
\[a(b + c) = ab + ac\]

• Apply this property to expand expressions and combine like terms.
• When multiplying binomials, use it to systematically distribute each term.

By using the distributive property during rationalization, you expand both the numerator and the denominator separately.
This process ensures that all roots are managed and simplified efficiently. Understanding this property is crucial when working with any algebraic expressions.

Simplifying Expressions

Simplifying expressions is often the final step after applying various algebraic techniques like conjugates and distributive property.
The goal is to arrive at the most straightforward form without any unnecessary complexities such as square roots in the denominator. Here's how you simplify:

• Combine and simplify terms that are similar or like each other.
• Cancel out terms wherever possible, particularly those that appear in both the numerator and denominator.

Finally, ensure that the expression reads clearly and logically.
Simplification might seem mundane, but it's essential for presenting clean and precise solutions, whether in textbooks or in practical applications.
With practice, identifying and combining terms becomes second nature, leading to efficient solution processes.