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$$\sqrt{2^5}$$

Step-by-step explanation:

First, deal with the product of the two powers of 2 inside the parentheses.

The two factors are powers of 2, so add the exponents.

$$(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}})^2 =$$

$$= (2^{\frac{1}{2} + \frac{3}{4}})^2$$

You need a common denominator, 4, to add the fractions.

$$= (2^{\frac{2}{4} + \frac{3}{4}})^2$$

$$= (2^{\frac{5}{4}})^2$$

Now you have an exponent raised to an exponent. Multiply the exponents and reduce the fraction.

$$= 2^{\frac{5}{4} \times 2}$$

$$= 2^{\frac{10}{4}}$$

$$= 2^{\frac{5}{2}}$$

When a fraction is an exponent, the numerator is an exponent and the denominator is the index of the root. A denominator of 2 means a root index of 2 which means square root.

$$= \sqrt{2^5}$$
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