6-3√6/9-√6 rationalize the denominator

To rationalize the denominator of the expression:

\[
\frac{6 - 3\sqrt{6}}{9 - \sqrt{6}}
\]

we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(9 - \sqrt{6}\) is \(9 + \sqrt{6}\).

### Step-by-Step Solution

1. **Multiply the numerator and denominator by the conjugate of the denominator:**

   \[
   \frac{6 - 3\sqrt{6}}{9 - \sqrt{6}} \times \frac{9 + \sqrt{6}}{9 + \sqrt{6}}
   \]

   This gives us:

   \[
   \frac{(6 - 3\sqrt{6})(9 + \sqrt{6})}{(9 - \sqrt{6})(9 + \sqrt{6})}
   \]

2. **Simplify the Denominator:**

   The denominator is now a difference of squares:

   \[
   (9 - \sqrt{6})(9 + \sqrt{6}) = 9^2 - (\sqrt{6})^2 = 81 - 6 = 75
   \]

   So, the denominator becomes \(75\).

3. **Expand the Numerator:**

   Using the distributive property on the numerator:

   \[
   (6 - 3\sqrt{6})(9 + \sqrt{6}) = 6 \cdot 9 + 6 \cdot \sqrt{6} - 3\sqrt{6} \cdot 9 - 3\sqrt{6} \cdot \sqrt{6}
   \]

   Calculating each term:

   - \(6 \cdot 9 = 54\)
   - \(6 \cdot \sqrt{6} = 6\sqrt{6}\)
   - \(-3\sqrt{6} \cdot 9 = -27\sqrt{6}\)
   - \(-3\sqrt{6} \cdot \sqrt{6} = -3 \cdot 6 = -18\)

   Combining these, the numerator becomes:

   \[
   54 + 6\sqrt{6} - 27\sqrt{6} - 18 = (54 - 18) + (6\sqrt{6} - 27\sqrt{6})
   \]

   Simplify further:

   \[
   = 36 - 21\sqrt{6}
   \]

4. **Combine the Result:**

   Now we have:

   \[
   \frac{36 - 21\sqrt{6}}{75}
   \]

5. **Simplify the Fraction:**

   We can divide both terms in

6 - 3√6 / 9 - √6 rationalize the denominator

Explore the step-by-step process of rationalizing the denominator of the expression 6 - 3√6 / 9 - √6 by using conjugates and the difference of squares formula. This solution covers key algebraic concepts suitable for students in Grades 9-10.

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