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Let's solve the given inequality:

\[ 2^{x^2 - 2x} < \sqrt{4^{x - p}} \]

We know from the properties of exponents and square roots that:

\[
\sqrt{4^{x-p}} = 2^{\frac{1}{2} \cdot 2(x-p)} = 2^{x-p}
\]

Thus, the inequality can be rewritten as:

\[
2^{x^2 - 2x} < 2^{x-p}
\]

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

\[
x^2 - 2x < x - p
\]

Rearrange the inequality:

\[
x^2 - 3x + p < 0
\]

This inequality represents a quadratic expression in terms of \(x\). The quadratic inequality will hold for the values of \(x\) within the range where the quadratic expression is negative. Given that \(1 < x < 2\), we need to find the value of \(p\) such that the quadratic expression \(x^2 - 3x + p\) is negative for this range of \(x\).

The roots of the quadratic equation \(x^2 - 3x + p = 0\) are given by:

\[
x = \frac{3 \pm \sqrt{9 - 4p}}{2}
\]

For the quadratic expression \(x^2 - 3x + p\) to be negative between \(1 < x < 2\), the roots of the equation should be real, and one root should be less than 1, and the other root should be greater than 2. This requires the discriminant to be non-negative:

\[
9 - 4p \geq 0
\]

\[
p \leq \frac{9}{4} = 2.25
\]

Thus, \(p\) must be less than or equal to 2.25, and by checking the specific options provided, we can see which value satisfies the inequality over the entire interval \(1 < x < 2\).

Let's check \(p = 2\) (since it's closest to 2.25):

\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]

This quadratic expression is negative between \(1 < x < 2\). Therefore, the correct answer is:

**b. 2**

Would you like more details or have any questions? Here are five related questions:

1. What is the general method for solving quadratic inequalities?
2. How do you determine the sign of a quadratic expression in a given interval?
3. Why is it important to check the range \(1 < x < 2\) for this problem?
4. How does the discriminant help in solving quadratic inequalities?
5. What are the implications if \(p > 2.25\) in this problem?

**Tip:** When solving quadratic inequalities, sketching the parabola can help visualize the intervals where the inequality holds.

Learn how to solve the inequality involving exponents and quadratic expressions: 2^(x^2 - 2x) < sqrt(4^(x - p)). This problem covers advanced concepts in exponents, inequalities, and quadratic equations. Suitable for advanced high school students and beyond.

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