$$(\sqrt[3]{27})^2 = $$

Correct.
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Convert the root to a fractional power (the power is the top of the fraction; the root is the bottom of the fraction):
>$$\displaystyle (\sqrt[3]{27})^2 = 27^{\frac{2}{3}}$$.
Rewrite 27 as a power of 3:
>$$\displaystyle 27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}}$$.
Now multiply the exponents and reduce. Remember the rule for a power to an exponent: $$(a^m)^n = a^{mn}$$.
>$$\displaystyle (3^3)^{\frac{2}{3}} = 3^{3\times \frac{2}{3}} = 3^2 = 9$$.

Incorrect.
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Incorrect.
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Incorrect.
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$$\sqrt{3}$$

$$3$$

$$9$$

$$\sqrt{27}$$