https://file.aiquickdraw.com/mathbot/file/17372140508556l6eimpp.jpg

Let's solve the given problems from the uploaded image. 

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**Problem (a):**  
\[
\frac{2}{1 - \sqrt{2}}
\]

To rationalize the denominator, multiply both numerator and denominator by the conjugate of \(1 - \sqrt{2}\), which is \(1 + \sqrt{2}\):

\[
\frac{2}{1 - \sqrt{2}} \cdot \frac{1 + \sqrt{2}}{1 + \sqrt{2}} = \frac{2(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})}
\]

Simplify the denominator using the difference of squares:

\[
(1 - \sqrt{2})(1 + \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1
\]

Now simplify the fraction:

\[
\frac{2(1 + \sqrt{2})}{-1} = -2(1 + \sqrt{2}) = -2 - 2\sqrt{2}
\]

**Final Answer (a):**
\[
-2 - 2\sqrt{2}
\]

---

**Problem (b):**  
\[
\frac{z - 9}{3 - \sqrt{z}}
\]

To rationalize the denominator, multiply both numerator and denominator by the conjugate of \(3 - \sqrt{z}\), which is \(3 + \sqrt{z}\):

\[
\frac{z - 9}{3 - \sqrt{z}} \cdot \frac{3 + \sqrt{z}}{3 + \sqrt{z}} = \frac{(z - 9)(3 + \sqrt{z})}{(3 - \sqrt{z})(3 + \sqrt{z})}
\]

Simplify the denominator using the difference of squares:

\[
(3 - \sqrt{z})(3 + \sqrt{z}) = 3^2 - (\sqrt{z})^2 = 9 - z
\]

So, the expression becomes:

\[
\frac{(z - 9)(3 + \sqrt{z})}{9 - z}
\]

Factor out \(-(z - 9)\) from the denominator:

\[
9 - z = -(z - 9)
\]

Substitute back:

\[
\frac{(z - 9)(3 + \sqrt{z})}{-(z - 9)} = -(3 + \sqrt{z})
\]

**Final Answer (b):**
\[
-3 - \sqrt{z}
\]

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### Let me know if you'd like further clarification or details.  
Here are **five related questions** to practice:
1. Rationalize \(\frac{3}{2 - \sqrt{3}}\).
2. Simplify \(\frac{5 + \sqrt{7}}{2 - \sqrt{7}}\) by rationalizing the denominator.
3. Rationalize \(\frac{1}{\sqrt{5} - 1}\).
4. Prove the general formula for rationalizing \( \frac{1}{a - \sqrt{b}} \).
5. Simplify \(\frac{4z - 1}{3 + \sqrt{z}}\) by rationalizing.

**Tip:** When rationalizing denominators, always look for the conjugate to simplify expressions involving square roots!

For each expression, please rationalize the denominator (in a & b) or numerator (in c & d) and simplify.

Learn how to rationalize the denominators in radical expressions like 2/(1-√2) and (z-9)/(3-√z) with detailed step-by-step solutions. This guide covers algebraic techniques, the difference of squares, and the conjugate method. Ideal for Grades 9-11.

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