Problem 71 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\( \frac{x - 8\sqrt{x} + 12}{x - 36} \) is the rationalized expression.

Step 1: Identify the Conjugate

To rationalize the denominator, we need to eliminate the square root from the denominator. To do this, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{x} + 6 \) is \( \sqrt{x} - 6 \).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply the entire expression \( \frac{\sqrt{x} - 2}{\sqrt{x} + 6} \) by \( \frac{\sqrt{x} - 6}{\sqrt{x} - 6} \). This will yield:\[ \frac{(\sqrt{x} - 2)(\sqrt{x} - 6)}{(\sqrt{x} + 6)(\sqrt{x} - 6)} \]

Step 3: Simplify the Denominator Using the Difference of Squares

Apply the difference of squares formula \((a + b)(a - b) = a^2 - b^2\) to the denominator:\[ (\sqrt{x})^2 - 6^2 = x - 36 \]

Step 4: Expand and Simplify the Numerator

Expand the numerator \((\sqrt{x} - 2)(\sqrt{x} - 6)\):\[ \sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot 6 - 2 \cdot \sqrt{x} + 2 \cdot 6 = x - 6\sqrt{x} - 2\sqrt{x} + 12 \]Combine like terms:\[ x - 8\sqrt{x} + 12 \]

Step 5: Write the Final Expression

Now that both the numerator and the denominator are simplified, the rationalized expression becomes:\[ \frac{x - 8\sqrt{x} + 12}{x - 36} \]

Key Concepts

Conjugate Method

The conjugate method is a clever technique used in mathematics to simplify expressions, especially when dealing with square roots in the denominator. Imagine you have a fraction with a square root in its denominator, like \(\frac{a}{\sqrt{b}+c}\). To "rationalize" this expression, we use the conjugate.

• Definition: The conjugate of a binomial \(a + b\) is \(a - b\).
• Process: Multiply both the numerator and the denominator by this conjugate to remove the square root from the denominator.

For our problem, the fraction is \(\frac{\sqrt{x}-2}{\sqrt{x}+6}\). Here, the conjugate of \(\sqrt{x}+6\) is \(\sqrt{x}-6\). By multiplying both the numerator and denominator by this conjugate, we remove the square root from the denominator. This step is crucial because it sets the stage for further simplification, making our expressions more manageable.

Difference of Squares

The difference of squares is a special algebraic identity. The formula \( (a+b)(a-b) = a^2 - b^2 \) helps to simplify products of binomials.

• Why It Works: When two terms are 'conjugates' of each other, multiplying them results in getting rid of the middle terms, simplifying the expression to a difference of two squares.
• Application: This is particularly useful in rationalizing denominators, hence its frequent use in conjunction with the conjugate method.

In our example, when we multiply \( (\sqrt{x}+6)(\sqrt{x}-6) \), we apply the difference of squares:
\[(\sqrt{x})^2 - 6^2 = x - 36\]This transformation results in a simpler expression without square roots in the denominator, which is both easier to work with and better for further algebraic manipulations.

Simplifying Expressions

Simplifying expressions involves reducing a mathematical expression to its simplest form, ensuring easier computation and clearer insight into the nature of the relationship between the variables and constants.

• Expand and Combine: By expanding the terms and combining like terms, we reduce complexity.
• Numerator Focus: In our scenario, after rationalizing the denominator, the work continues with simplifying the numerator.

For the numerator \((\sqrt{x}-2)(\sqrt{x}-6)\), the steps: expand using the distributive property, then simplify:
\[\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot 6 - 2 \cdot \sqrt{x} + 2 \cdot 6 = x - 8\sqrt{x} + 12 \]This expression is then combined to provide the simplified form.
Concluding, these processes enhance our ability to interpret and manipulate algebraic expressions articulately, allowing them to present clearer, more intuitive relationships.