We are given the equation:

\[
(\sqrt{3} - 1)^{30} = (4 - 2\sqrt{3})^{2x - 3}
\]

### Step 1: Simplify \( 4 - 2\sqrt{3} \)
Notice that:

\[
4 - 2\sqrt{3} = (\sqrt{3} - 1)^2
\]

We can verify this by expanding \( (\sqrt{3} - 1)^2 \):

\[
(\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2(\sqrt{3})(1) + 1^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}
\]

Thus:

\[
4 - 2\sqrt{3} = (\sqrt{3} - 1)^2
\]

### Step 2: Substitute this back into the original equation

We now have:

\[
(\sqrt{3} - 1)^{30} = \left( (\sqrt{3} - 1)^2 \right)^{2x - 3}
\]

Simplifying the right-hand side:

\[
(\sqrt{3} - 1)^{30} = (\sqrt{3} - 1)^{2(2x - 3)} = (\sqrt{3} - 1)^{4x - 6}
\]

### Step 3: Set the exponents equal to each other

Since the bases are the same, we can equate the exponents:

\[
30 = 4x - 6
\]

### Step 4: Solve for \( x \)

Add 6 to both sides:

\[
30 + 6 = 4x
\]

\[
36 = 4x
\]

Divide by 4:

\[
x = 9
\]

Thus, the solution is:

\[
x = 9
\]

Let me know if you'd like further details or have any questions!

Here are 5 related questions to explore further:
1. How would the result change if we changed \( (\sqrt{3} - 1) \) to \( (\sqrt{2} - 1) \)?
2. What if the exponents were fractional instead of whole numbers?
3. How do you generalize this technique for higher powers of binomial terms?
4. Can this technique be used to solve logarithmic equations involving radicals?
5. How would the solution change if the right-hand side were \( (4 + 2\sqrt{3}) \)?

**Tip:** Always verify your work by plugging the solution back into the original equation to ensure both sides are equal.

Discover how to solve the exponential equation involving radicals: (√3 - 1)^30 = (4 - 2√3)^(2x - 3). Learn how to simplify the equation and find the solution step-by-step, covering key concepts such as radicals, exponentiation, and algebra. Suitable for students in Grades 10-12.

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