Problem 109 - College Algebra with Modeling and Visualization

Question

Rationalize the denominator. $$ \frac{\sqrt{z}}{\sqrt{z}-3} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{z + 3\sqrt{z}}{z - 9} \).

Step 1: Identify the Conjugate

To rationalize the denominator, identify the conjugate of the denominator \( \sqrt{z} - 3 \). The conjugate is \( \sqrt{z} + 3 \).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( \sqrt{z} + 3 \). This gives:\[ \frac{\sqrt{z}}{\sqrt{z} - 3} \times \frac{\sqrt{z} + 3}{\sqrt{z} + 3} = \frac{\sqrt{z}(\sqrt{z} + 3)}{(\sqrt{z} - 3)(\sqrt{z} + 3)} \]

Step 3: Simplify the Numerator

Distribute \( \sqrt{z} \) in the numerator:\[ \sqrt{z}(\sqrt{z} + 3) = z + 3\sqrt{z} \]

Step 4: Simplify the Denominator

Apply the difference of squares formula to the denominator:\[ (\sqrt{z} - 3)(\sqrt{z} + 3) = (\sqrt{z})^2 - (3)^2 = z - 9 \]

Step 5: Combine and Simplify

Combine the simplified numerator and denominator:\[ \frac{z + 3\sqrt{z}}{z - 9} \]

Key Concepts

Conjugate in Algebra

In algebra, the conjugate of an expression is a very useful tool, especially when dealing with radicals. The conjugate of a binomial expression like \( \sqrt{z} - 3 \) is formed by changing the sign of the second term. Therefore, its conjugate becomes \( \sqrt{z} + 3 \). By multiplying an expression by its conjugate, we can eliminate radicals or complex numbers from denominators. This process is crucial for simplifying expressions and is often used to rationalize denominators. When dealing with a fraction, if the denominator contains a radical, multiplying both the numerator and the denominator by the conjugate helps to "cancel out" the radical in the denominator, making the expression more rational and easy to understand.

Difference of Squares

The difference of squares is a mathematical identity or pattern used to simplify expressions. It is defined by the equation \((a^2 - b^2 = (a-b)(a+b)\). This formula helps simplify expressions that follow the structure of a squared value minus another squared value.

With the example of rationalizing the denominator from earlier:

• Apply \((\sqrt{z} - 3)(\sqrt{z} + 3)\)
• This becomes \((\sqrt{z})^2 - (3)^2 = z - 9\)

This formula efficiently simplifies the denominator, leaving it with no radical. This step is crucial in rationalizing the entire expression, as it allows you to re-write the expression in a simpler, more easily understandable form.

Simplifying Radicals

Simplifying radicals is about expressing radical expressions in their simplest form. Radicals can often be seen in algebra and mathematics, but when left unsimplified, they can make expressions confusing or needlessly complex. Simplifying radicals involves a few steps:

• Factor the expression to find perfect squares within the radical
• Remove these squares by calculating their square roots
• Simplify any remaining radicals

In the process of rationalizing \(\frac{\sqrt{z}}{\sqrt{z} - 3}\), simplifying radicals isn't particularly aimed at the numerator. However, understanding simplification is essential in algebra. By simplifying radicals, we often achieve nicer expressions, especially when later used in operations like addition or subtraction, where terms need to be more coherent and manageable. Moreover, clear expression simplification aids in recognizing patterns and making further algebraic manipulations more feasible.