Problem 112 - College Algebra with Modeling and Visualization

Question

Rationalize the denominator. $$ \frac{1}{\sqrt{a+1}+\sqrt{a}} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \sqrt{a+1} - \sqrt{a} \).

Step 1: Identify the Expression to Rationalize

The given expression is \( \frac{1}{\sqrt{a+1}+\sqrt{a}} \). The goal is to eliminate the square roots in the denominator.

Step 2: Determine the Conjugate

The conjugate of \( \sqrt{a+1} + \sqrt{a} \) is \( \sqrt{a+1} - \sqrt{a} \). Multiplying by the conjugate will eliminate the radicals in the denominator.

Step 3: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( \sqrt{a+1} - \sqrt{a} \):\[ \frac{1}{\sqrt{a+1}+\sqrt{a}} \times \frac{\sqrt{a+1}-\sqrt{a}}{\sqrt{a+1}-\sqrt{a}} = \frac{\sqrt{a+1}-\sqrt{a}}{(\sqrt{a+1}+\sqrt{a})(\sqrt{a+1}-\sqrt{a})} \]

Step 4: Simplify the Denominator Using the Difference of Squares

The product \((\sqrt{a+1}+\sqrt{a})(\sqrt{a+1}-\sqrt{a})\) simplifies using the difference of squares formula:\[ (\sqrt{a+1})^2 - (\sqrt{a})^2 = (a+1) - a = 1 \]

Step 5: Write the Rationalized Expression

Now that the denominator is simplified to 1, the expression becomes:\[ \frac{\sqrt{a+1}-\sqrt{a}}{1} = \sqrt{a+1} - \sqrt{a} \]

Key Concepts

Conjugate

Calculating the conjugate in mathematical operations is a nifty trick used to rationalize denominators.This technique is particularly helpful when dealing with square roots or radicals. In our given expression, \( \frac{1}{\sqrt{a+1}+\sqrt{a}} \), the conjugate is \( \sqrt{a+1} - \sqrt{a} \). The concept here is to "pair up" the expression. By multiplying the original expression by its conjugate over itself, we essentially multiply by 1, as \( \frac{\sqrt{a+1} - \sqrt{a}}{\sqrt{a+1} - \sqrt{a}} = 1 \).

• This keeps the value of the expression the same.
• It cleverly alters the form to remove radicals from the denominator.

Thus, the conjugate trick is a fantastic way to simplify calculations and neaten up your solutions!

Difference of Squares

The technique of using the difference of squares provides a compact route to simplify expressions with radicals. This concept is embedded in algebra and involves the identity: \((x + y)(x - y) = x^2 - y^2\). In the solution, when both the numerator and denominator are multiplied by \( \sqrt{a+1} - \sqrt{a} \), applying the difference of squares formula is key to simplify the denominator. By multiplying \((\sqrt{a+1} + \sqrt{a})(\sqrt{a+1} - \sqrt{a})\), we have:\[ (otele<|meta_end|>the blank space here. \sqrt{a+1})^2 - (\sqrt{a})^2 = (a+1) - a = 1 \].

• This cleverly eliminates the radicals in the denominator, turning it into a simpler whole number.
• Simplification occurs seamlessly.

This approach makes rationalizing, which might initially seem complicated, a straightforward and manageable process!

Simplifying Radicals

Simplifying radicals is all about transforming radical expressions into a more manageable form. Radicals often appear daunting, but breaking them down can make calculations much easier. In the initial expression \( \frac{1}{\sqrt{a+1}+\sqrt{a}} \), we aimed to remove the square roots from the denominator.

• By working on simplifying the denominator using the conjugate, we effectively remove these radicals.
• Simplification often results in a numerical denominator, as seen in the expression: \(1\).

Now, look at the numerator. After simplification, it becomes \( \sqrt{a+1} - \sqrt{a} \). This step-by-step approach ensures the final expression is neat and devoid of any radicals in the denominator.In essence, simplifying radicals is about transforming radical expressions to a form that's easier to work with, making math less tricky and more fun!