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### 4. Simplify \(\frac{5 - \sqrt{3}}{2 + \sqrt{3}}\)
We need to simplify this expression and express the result in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers.

To simplify, we multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the square root:

\[
\frac{5 - \sqrt{3}}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{(5 - \sqrt{3})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}
\]

Let's calculate this step by step:
1. The denominator becomes:
   \[
   (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1
   \]
2. The numerator becomes:
   \[
   (5 - \sqrt{3})(2 - \sqrt{3}) = 5(2) - 5(\sqrt{3}) - \sqrt{3}(2) + \sqrt{3}\cdot\sqrt{3}
   = 10 - 5\sqrt{3} - 2\sqrt{3} + 3 = 13 - 7\sqrt{3}
   \]

So, the expression simplifies to:
\[
\frac{13 - 7\sqrt{3}}{1} = 13 - 7\sqrt{3}
\]

Thus, the simplified expression is:
\[
\boxed{13 - 7\sqrt{3}}
\]
Where \(a = 13\) and \(b = -7\).

---

### 6. Solve the equation \(2^{(1-x)} = 4^x\)

We rewrite \(4^x\) as a power of 2:
\[
4^x = (2^2)^x = 2^{2x}
\]

Now, the equation becomes:
\[
2^{1-x} = 2^{2x}
\]

Since the bases are the same, we can equate the exponents:
\[
1 - x = 2x
\]

Solving for \(x\):
\[
1 = 3x \quad \Rightarrow \quad x = \frac{1}{3}
\]

Thus, the solution is:
\[
\boxed{x = \frac{1}{3}}
\]

---

Do you want more details on any of the steps or have further questions?

Here are some related questions you might want to explore:
1. How do you simplify expressions with square roots using the conjugate method?
2. How can you solve more complicated exponential equations?
3. What are some real-life applications of simplifying radical expressions?
4. What are the properties of exponents that help in solving exponential equations?
5. How do you handle equations where the bases are not the same in exponential form?

**Tip:** Always check if the denominator can be simplified by multiplying by the conjugate when working with radicals.

Learn how to simplify the expression (5 - √3) / (2 + √3) and solve the exponential equation 2^(1-x) = 4^x. This problem covers key algebraic concepts including rationalizing the denominator and properties of exponents. Suitable for students in Grades 9-12.

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