Problem 34 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{\sqrt{a}}{\sqrt{a}-2}\)

Step-by-Step Solution

Answer

The simplest form is \( \frac{a + 2\sqrt{a}}{a - 4} \).

Step 1: Understand the Problem

We need to rationalize the denominator of the fraction \( \frac{\sqrt{a}}{\sqrt{a}-2} \). This means eliminating the square root from the denominator.

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate of the denominator \( \sqrt{a}-2 \), which is \( \sqrt{a}+2 \). This gives us: \[ \frac{\sqrt{a} \cdot (\sqrt{a}+2)}{(\sqrt{a}-2)(\sqrt{a}+2)} \].

Step 3: Apply the Difference of Squares Formula

The denominator \((\sqrt{a}-2)(\sqrt{a}+2)\) represents a difference of squares, which simplifies to: \( (\sqrt{a})^2 - 2^2 = a - 4 \).

Step 4: Simplify the Numerator

Expand the numerator: \( \sqrt{a} \cdot (\sqrt{a}+2) = \sqrt{a} \cdot \sqrt{a} + \sqrt{a} \cdot 2 = a + 2\sqrt{a} \).

Step 5: Write the Simplified Expression

Combine the results from the previous steps to get the final expression: \[ \frac{a + 2\sqrt{a}}{a - 4} \]. This cannot be simplified further since \(a + 2\sqrt{a}\) and \(a - 4\) have no common factors.

Key Concepts

Difference of Squares

The concept of the difference of squares is a nifty algebraic technique used to simplify expressions involving two terms that are squared and subtracted from each other. It's represented by the formula:

• \((x - y)(x + y) = x^2 - y^2\)

In our original exercise, the denominator was expressed as \((\sqrt{a} - 2)(\sqrt{a} + 2)\). This setup is perfect for applying the difference of squares formula. When we apply it, we essentially multiply these expressions to get a simpler form, eliminating the square root:

• \((\sqrt{a})^2 - 2^2 = a - 4\)

Using this rule helps us simplify complex expressions, and especially helps in rationalizing the denominator, which is transforming it into a rational number without radicals.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves making an expression as simple as possible. In the context of rationalizing the denominator, it means manipulating an expression so the roots are removed from the denominator, making it easier to work with in subsequent calculations.

After applying the difference of squares to simplify the denominator in the original problem, we also need to simplify the numerator. Let's look at how we do this:

• First, multiply the terms in the numerator: \(\sqrt{a} \times (\sqrt{a} + 2)\). This gives us:
  • \(\sqrt{a} \times \sqrt{a} = a\)
  • \(\sqrt{a} \times 2 = 2\sqrt{a}\)
  Thus the numerator becomes \(a + 2\sqrt{a}\).

At the end, ensure all expressions are in their simplest form, meaning they can't be reduced further. Recognizing when an expression cannot be simplified further is vital in ensuring clarity and accuracy.

Conjugate Multiplication

Conjugate multiplication is an essential method used to simplify fractions involving roots. A conjugate in this context refers to altering the sign between two terms. For our given exercise, the denominator \(\sqrt{a} - 2\) has a conjugate \(\sqrt{a} + 2\). Here's why it's useful:

• When you multiply an expression by its conjugate, it helps eliminate the radicals in the denominator through the difference of squares.
• Multiplying both the numerator and denominator by the conjugate, keeps the fraction equivalent to the original.

This results in a new fraction that has a rational number in the denominator:

• Original expression: \(\frac{\sqrt{a}}{\sqrt{a} - 2}\)
• After conjugate multiplication: \(\frac{\sqrt{a}(\sqrt{a} + 2)}{(\sqrt{a} - 2)(\sqrt{a} + 2)} = \frac{a + 2\sqrt{a}}{a - 4}\)

Once the denominator becomes a rational number, calculations become more straightforward, which is the main goal of this method.