Simplify the following expression.
5
𝑥𝑥 + 𝑏𝑏 − 5
𝑥𝑥
𝑏𝑏
4. Rationalize the numerator.
√𝑥𝑥 + 6 − √𝑎𝑎 + 6
𝑥𝑥 − 𝑎𝑎

Let's simplify and rationalize the expression step by step.

### 1. Simplify the Expression \( \frac{5x + b - \frac{5x}{b}}{4} \)

First, let's consider the expression inside the fraction.

\[
5x + b - \frac{5x}{b}
\]

To combine these terms, find a common denominator for the terms involving \(x\):

\[
5x + \frac{b^2}{b} - \frac{5x}{b} = \frac{5xb + b^2 - 5x}{b}
\]

Simplifying further:

\[
\frac{(5x)(b) - (5x) + b^2}{b} = \frac{5x(b - 1) + b^2}{b}
\]

Now, divide the entire expression by 4:

\[
\frac{\frac{5x(b - 1) + b^2}{b}}{4} = \frac{5x(b - 1) + b^2}{4b}
\]

This is the simplified expression.

### 2. Rationalize the Numerator \( \frac{\sqrt{x + 6} - \sqrt{a + 6}}{x - a} \)

To rationalize the numerator, multiply both the numerator and the denominator by the conjugate of the numerator:

\[
\text{Conjugate of the numerator: } \sqrt{x + 6} + \sqrt{a + 6}
\]

\[
\frac{\sqrt{x + 6} - \sqrt{a + 6}}{x - a} \times \frac{\sqrt{x + 6} + \sqrt{a + 6}}{\sqrt{x + 6} + \sqrt{a + 6}} = \frac{(\sqrt{x + 6})^2 - (\sqrt{a + 6})^2}{(x - a)(\sqrt{x + 6} + \sqrt{a + 6})}
\]

Simplify the numerator:

\[
(\sqrt{x + 6})^2 - (\sqrt{a + 6})^2 = (x + 6) - (a + 6) = x - a
\]

Thus, the expression becomes:

\[
\frac{x - a}{(x - a)(\sqrt{x + 6} + \sqrt{a + 6})}
\]

Cancel out the \(x - a\) terms:

\[
\frac{1}{\sqrt{x + 6} + \sqrt{a + 6}}
\]

This is the rationalized form of the expression.

### Final Results:
1. The simplified expression is \(\frac{5x(b - 1) + b^2}{4b}\).
2. The rationalized expression is \(\frac{1}{\sqrt{x + 6} + \sqrt{a + 6}}\).

Would you like more details on these steps or explanations on specific parts?

### Related Questions:
1. How would the simplification change if \(b = 0\)?
2. What happens when \(b = 1\) in the simplified expression?
3. How does rationalizing the denominator differ from rationalizing the numerator?
4. What is the significance of the conjugate in rationalizing expressions?
5. How can the process of simplification help in solving algebraic equations?

### Tip:
When simplifying complex algebraic expressions, it's often helpful to look for patterns such as common factors or conjugates that can simplify the process significantly.

Learn how to simplify the expression 5x + b - 5x / b and rationalize the numerator √x + 6 - √a + 6 / x - a step-by-step. This detailed guide covers algebraic concepts including fractional expression simplification and rationalization. Suitable for high school students studying algebra.

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