https://file.aiquickdraw.com/mathbot/file/172670760779078ngl5zn.jpg

The expression in the image appears to be:

\[
\frac{4\sqrt{5}}{3 - 2\sqrt{5}}
\]

This is a fraction with a radical in the denominator, which can be simplified by rationalizing the denominator. To do this, we need to multiply both the numerator and denominator by the conjugate of the denominator, \( 3 + 2\sqrt{5} \).

Let's go through the steps:

### Step 1: Multiply the numerator and denominator by the conjugate of the denominator
We multiply both by \( 3 + 2\sqrt{5} \):

\[
\frac{4\sqrt{5}}{3 - 2\sqrt{5}} \times \frac{3 + 2\sqrt{5}}{3 + 2\sqrt{5}}
\]

### Step 2: Apply the difference of squares in the denominator
Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we simplify the denominator:

\[
(3 - 2\sqrt{5})(3 + 2\sqrt{5}) = 3^2 - (2\sqrt{5})^2 = 9 - 20 = -11
\]

### Step 3: Simplify the numerator
Now, multiply the numerators:

\[
4\sqrt{5} \times (3 + 2\sqrt{5}) = 4\sqrt{5} \times 3 + 4\sqrt{5} \times 2\sqrt{5} = 12\sqrt{5} + 8 \times 5 = 12\sqrt{5} + 40
\]

### Step 4: Put it all together
The expression becomes:

\[
\frac{12\sqrt{5} + 40}{-11}
\]

This simplifies to:

\[
-\frac{12\sqrt{5} + 40}{11}
\]

Thus, the final answer is:

\[
-\frac{12\sqrt{5} + 40}{11}
\]

Would you like further clarification or details on any of these steps?

Here are some related questions to consider:
1. How do you rationalize denominators with different radicals?
2. What is the conjugate of a binomial expression involving radicals?
3. How does the difference of squares identity work with radicals?
4. Can you apply this method to more complex radical expressions?
5. How can you simplify expressions with nested radicals?

**Tip**: Always check if the denominator contains a radical before simplifying a fraction to avoid any unnecessary complexity in the final answer.

This problem explains how to rationalize the denominator in an expression with radicals. Learn step-by-step how to simplify \(\frac{4\sqrt{5}}{3 - 2\sqrt{5}}\) by multiplying by the conjugate and applying the difference of squares theorem. Suitable for Grades 9-11.

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