1. **State the problem:** Solve the equation $$(2x)^{\frac{3}{4}} + 2 = 10$$ for $x$. 2. **Isolate the term with the exponent:** Subtract 2 from both sides: $$ (2x)^{\frac{3}{4}} + 2 - 2 = 10 - 2 $$ $$ (2x)^{\frac{3}{4}} = 8 $$ 3. **Remove the fractional exponent:** To undo the exponent $\frac{3}{4}$, raise both sides to the reciprocal power $\frac{4}{3}$: $$ \left((2x)^{\frac{3}{4}}\right)^{\frac{4}{3}} = 8^{\frac{4}{3}} $$ 4. **Simplify the left side:** Using the power of a power rule $$(a^{m})^{n} = a^{mn}$$: $$ (2x)^{\frac{3}{4} \times \frac{4}{3}} = (2x)^1 = 2x $$ 5. **Simplify the right side:** Calculate $8^{\frac{4}{3}}$. First, write 8 as $2^3$: $$ 8^{\frac{4}{3}} = (2^3)^{\frac{4}{3}} = 2^{3 \times \frac{4}{3}} = 2^4 = 16 $$ 6. **Now the equation is:** $$ 2x = 16 $$ 7. **Solve for $x$ by dividing both sides by 2:** $$ x = \frac{16}{2} $$ $$ x = 8 $$ **Final answer:** $$ \boxed{8} $$