1. The problem asks to find an expression equivalent to $2q^{3/4}$. 2. Recall that rational exponents and radicals are related by the rule: $$a^{m/n} = \sqrt[n]{a^m}$$ where $m$ is the power and $n$ is the root. 3. Applying this to $q^{3/4}$, we get: $$q^{3/4} = \sqrt[4]{q^3}$$ 4. Therefore, $2q^{3/4}$ can be written as: $$2 \times \sqrt[4]{q^3}$$ 5. Now, let's check the options: - (1) $\sqrt[3]{2q^4}$ is the cube root, which corresponds to exponent $1/3$, not $3/4$. - (2) $2\sqrt[4]{q^3}$ matches exactly our expression. - (3) $\sqrt[4]{2q^3}$ is the fourth root of $2q^3$, which is $ (2q^3)^{1/4} = 2^{1/4} q^{3/4}$, not $2q^{3/4}$. - (4) $\frac{2^{3/4} \sqrt{q}}{\sqrt{q}} = 2^{3/4}$ after canceling $\sqrt{q}$, which is not equal to $2q^{3/4}$. 6. Hence, the equivalent expression is option (2): $$2 \sqrt[4]{q^3}$$ Final answer: $$2q^{3/4} = 2 \sqrt[4]{q^3}$$