4 - square root of x over 16x - x squared conjugate

To simplify the expression involving the conjugate, we first clarify the given expression:

\[
\frac{4 - \sqrt{x}}{16x - x^2}
\]

To rationalize the numerator or denominator, the conjugate of the numerator \(4 - \sqrt{x}\) is \(4 + \sqrt{x}\). Now, let's multiply both the numerator and denominator by this conjugate:

\[
\frac{(4 - \sqrt{x})(4 + \sqrt{x})}{(16x - x^2)(4 + \sqrt{x})}
\]

### Step-by-step process:

1. **Simplify the numerator:**
   \[
   (4 - \sqrt{x})(4 + \sqrt{x}) = 4^2 - (\sqrt{x})^2 = 16 - x
   \]
   So the numerator becomes \(16 - x\).

2. **Leave the denominator factored:**
   \[
   (16x - x^2)(4 + \sqrt{x})
   \]

Thus, the expression becomes:

\[
\frac{16 - x}{(16x - x^2)(4 + \sqrt{x})}
\]

This is the simplified form after multiplying by the conjugate.

Would you like more details or have any questions?

### Relative questions:
1. What is the importance of conjugates in simplifying expressions?
2. Can we further simplify this expression if specific values for \(x\) are given?
3. How do we determine when to use conjugates in algebraic fractions?
4. What happens if \(x\) is negative or irrational in this expression?
5. Can the denominator be factored further to simplify the entire expression?

### Tip:
Conjugates are particularly useful when you need to eliminate square roots from the numerator or denominator of a fraction.

Learn how to simplify algebraic fractions by using conjugates, focusing on the expression (4 - sqrt(x)) / (16x - x^2). This step-by-step guide breaks down the process of rationalization using the difference of squares theorem, perfect for high school students in Grades 9-12.

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