Problem 105 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \frac{3 \sqrt{2}-5 \sqrt{3}}{2 \sqrt{3}-3 \sqrt{2}} $$

Step-by-Step Solution

Answer

\(\frac{3 \sqrt{6}}{2} + 2\)

Step 1: Multiply the Numerator and Denominator by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 2 \sqrt{3} + 3 \sqrt{2} \). The expression becomes:\[\frac{(3 \sqrt{2} - 5 \sqrt{3})(2 \sqrt{3} + 3 \sqrt{2})}{(2 \sqrt{3} - 3 \sqrt{2})(2 \sqrt{3} + 3 \sqrt{2})}\]

Step 2: Expand the Denominator

Apply the difference of squares formula \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 2\sqrt{3}\) and \(b = 3\sqrt{2}\):\[(2 \sqrt{3})^2 - (3 \sqrt{2})^2 = 4 \cdot 3 - 9 \cdot 2 = 12 - 18 = -6\]Thus, the denominator simplifies to \(-6\).

Step 3: Expand the Numerator

Apply the distributive property to determine the expanded numerator:\[(3 \sqrt{2})(2 \sqrt{3}) + (3 \sqrt{2})(3 \sqrt{2}) - (5 \sqrt{3})(2 \sqrt{3}) - (5 \sqrt{3})(3 \sqrt{2})\]This simplifies to:\[ 6 \sqrt{6} + 9 \times 2 - 10 \times 3 - 15 \sqrt{6} \]Combine the terms:\[ -9 \sqrt{6} + 18 - 30 = -9 \sqrt{6} - 12\]

Step 4: Simplify the Expression

Combine the numerator and denominator from Steps 2 and 3:\[\frac{-9 \sqrt{6} - 12}{-6}\]Split into separate terms:\[\frac{-9 \sqrt{6}}{-6} + \frac{-12}{-6}\]This simplifies to:\[\frac{3 \sqrt{6}}{2} + 2\]

Step 5: Final Simplified Result

The radical expression is now fully simplified, and the answer is:\[\frac{3 \sqrt{6}}{2} + 2\]

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that include a root symbol, such as a square root or cube root, over numbers or variables. Understanding these expressions is crucial because they often appear in various algebraic problems. For example, in the expression \( \sqrt{3} \), 3 is what's under the radical sign, and the entire term represents its square root. Radical expressions can be combined through operations like addition, subtraction, multiplication, and division. However, one special rule when handling radicals is that you can't simply combine different radicands (the number beneath the root symbol) unless the radicands are identical. So, for \( 3\sqrt{2} + 5\sqrt{3} \), these terms can't be added directly together because the numbers under the radicals are different (2 and 3). However, when it comes to multiplication, the radicands can be multiplied together, thus \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This property becomes especially useful in problems involving the rationalization of denominators.

Simplifying Radicals

The process of simplifying radicals involves reducing the expression into its simplest form. It is often done by factoring the number inside the radical into its prime factors. For instance, \( \sqrt{18} \) can be expressed as \( \sqrt{2 \times 3^2} \). Since \( \sqrt{3^2} \) equals 3, the entire expression simplifies to \( 3\sqrt{2} \).

• This method helps in transforming complex expressions into simpler, equivalent expressions that are easier to work with.
• Simplifying radicals also simplifies practical calculations, particularly in solving algebraic fractions.

An important technique related to this is rationalizing denominators, which comes in handy when a radical appears in the denominator of a fraction. By eliminating radicals from the denominator, the resulting expression often becomes more straightforward and familiar.

Algebraic Fractions

Algebraic fractions are fractions in which the numerator, the denominator, or both, contain algebraic expressions. These could include variables, constants, and operators. Problems involving algebraic fractions often require the same treatment as any numerical fraction—they need to be simplified.

• To simplify, it's necessary to ensure no common factors exist between the numerator and the denominator.
• Another critical step involves rationalizing the denominator, making sure to multiply both the numerator and denominator by the conjugate of the denominator if any radicals are present.
• The goal is to make the denominator a rational number, simplifying the problem into an easier completed expression.

In practice, handling a problem with algebraic fractions is like navigating through a puzzle. You want to clear any obstacles, such as radicals in denominators, to make the expression more manageable and ready for further mathematical operations. Understanding how to manipulate these fractions appropriately is pivotal in developing strong algebraic skills.