1. **State the problem:** Calculate the value of $$4^2 - \left(\frac{2}{3}\right)^2 + \left(-\frac{3}{2}\right)^2 \times \frac{1}{3}$$. 2. **Recall the rules:** - Exponentiation means raising a number to a power. - Squaring a fraction means squaring both numerator and denominator. - Multiplication and subtraction follow order of operations (PEMDAS/BODMAS). 3. **Calculate each term:** - $$4^2 = 16$$ - $$\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$$ - $$\left(-\frac{3}{2}\right)^2 = \frac{(-3)^2}{2^2} = \frac{9}{4}$$ 4. **Multiply the last term by $$\frac{1}{3}$$:** $$\frac{9}{4} \times \frac{1}{3} = \frac{9 \times 1}{4 \times 3} = \frac{9}{12}$$ 5. **Simplify $$\frac{9}{12}$$ by canceling common factors:** $$\frac{\cancel{9}^3}{\cancel{12}^4} = \frac{3}{4}$$ 6. **Put it all together:** $$16 - \frac{4}{9} + \frac{3}{4}$$ 7. **Find common denominator for $$\frac{4}{9}$$ and $$\frac{3}{4}$$:** - Common denominator is $$36$$. - Convert fractions: $$\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}$$ $$\frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36}$$ 8. **Rewrite the expression:** $$16 - \frac{16}{36} + \frac{27}{36} = 16 + \frac{27 - 16}{36} = 16 + \frac{11}{36}$$ 9. **Convert 16 to fraction with denominator 36:** $$16 = \frac{16 \times 36}{36} = \frac{576}{36}$$ 10. **Add fractions:** $$\frac{576}{36} + \frac{11}{36} = \frac{576 + 11}{36} = \frac{587}{36}$$ **Final answer:** $$\boxed{\frac{587}{36}}$$ or approximately $$16.3056$$.