1. **State the problem:** Solve the equation $$\left(4y\right)^{\frac{1}{3}} + 3 = 5$$ for $y$. 2. **Isolate the cube root term:** Subtract 3 from both sides: $$\left(4y\right)^{\frac{1}{3}} = 5 - 3 = 2$$ 3. **Remove the cube root:** Cube both sides to eliminate the fractional exponent: $$\left(\left(4y\right)^{\frac{1}{3}}\right)^3 = 2^3$$ $$4y = 8$$ 4. **Solve for $y$:** Divide both sides by 4: $$y = \frac{8}{4} = 2$$ 5. **Check the solution:** Substitute $y=2$ back into the original equation: $$\left(4 \times 2\right)^{\frac{1}{3}} + 3 = \left(8\right)^{\frac{1}{3}} + 3 = 2 + 3 = 5$$ The solution satisfies the equation. **Final answer:** $$y = 2$$