Problem 61 - Algebra for College Students

Question

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{3}}{2 \sqrt{5}+4}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{15}}{2} - \sqrt{3}\)

Step 1: Identify the Conjugate

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The given denominator is \(2\sqrt{5} + 4\), and the conjugate is \(2\sqrt{5} - 4\).

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate \(2\sqrt{5} - 4\):\(\frac{\sqrt{3}}{2\sqrt{5} + 4} \cdot \frac{2\sqrt{5} - 4}{2\sqrt{5} - 4} = \frac{\sqrt{3}(2\sqrt{5} - 4)}{(2\sqrt{5} + 4)(2\sqrt{5} - 4)}\)

Step 3: Simplify the Denominator

The denominator is of the form \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 2\sqrt{5}\) and \(b = 4\). So,\((2\sqrt{5})^2 - 4^2 = 4 \cdot 5 - 16 = 20 - 16 = 4\).

Step 4: Simplify the Numerator

Distribute \(\sqrt{3}\) in the numerator:\[\sqrt{3}(2\sqrt{5} - 4) = 2\sqrt{3} \cdot \sqrt{5} - 4\sqrt{3} = 2\sqrt{15} - 4\sqrt{3}\]

Step 5: Combine the Results

Combine the simplified numerator and denominator:\[\frac{2\sqrt{15} - 4\sqrt{3}}{4}\]

Step 6: Simplify Further

Divide each term in the numerator by the denominator:\[\frac{2\sqrt{15}}{4} - \frac{4\sqrt{3}}{4} = \frac{\sqrt{15}}{2} - \sqrt{3}\]

Step 7: Final Step: Write the Simplified Expression

The expression is now fully simplified. The final result is:\[\frac{\sqrt{15}}{2} - \sqrt{3}\]

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating them. These symbols represent quantities without fixed values known as variables. In algebra, you can perform arithmetic operations just like with numbers. This is crucial when solving equations or working with expressions.
Algebra isn't just about finding the value of variables. It helps in formulating real-world problems to equations and finding solutions in a logical manner. For example, in the exercise provided, algebraic techniques are employed to manipulate terms to achieve a specific form, which is called 'rationalizing the denominator.' The goal is to eliminate any radicals or irrational numbers from the denominator using algebraic strategies.
It's important to remember the properties of operations in algebra, such as the distributive property which allows us to expand expressions like \(\sqrt{3}(2\sqrt{5} - 4)\). Mastery of these operations will enable you to maneuver through problems with ease and lead to more effective solutions.

Radical Expressions

Radical expressions are mathematical expressions that contain a square root, cube root, or any higher-order root. When dealing with radical expressions, it's vital to understand simplification processes, especially when they appear in fractions. This process often involves removing the radical from the denominator.
Manipulating these expressions requires understanding notations and properties of roots. For instance, the radical \(\sqrt{3}\) in the numerator of the exercise reflects a square root, which denotes a number that, when multiplied by itself, gives the original number inside the root. Simplification may involve combining like terms or rationalizing, as seen in the exercise where the initial task is to handle \(\sqrt{3}\) present in a fraction.

• Basic property: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\)
• To rationalize: Multiplies by a conjugate to eliminate the radical

Grasping these basic properties makes it easier to work through more complex algebraic manipulations when faced with radical expressions.

Conjugate Method

The conjugate method is a standard technique used in mathematics to rationalize denominators containing square roots. The method involves multiplying by a conjugate, which is formed by altering the sign between two terms in a binomial. In this exercise, the denominator \(2\sqrt{5} + 4\) has its conjugate \(2\sqrt{5} - 4\).
This method hinges on the identity \(a^2 - b^2 = (a+b)(a-b)\), known as the difference of squares, which allows us to eliminate the square root in the denominator when the multiplication is performed. The resulting denominator becomes a rational number, simplifying further calculations.
Here's an easy recap of the conjugate method process:

• Identify the binomial in the denominator.
• Switch the sign between terms to find the conjugate.
• Multiply both the numerator and denominator by this conjugate.
• Simplify using the difference of squares.

Using the conjugate is powerful because it transforms radicals to integers, making the fractions easier to work with and understand.