1. **State the problem:** Simplify the expression $$\frac{8}{5} - 2 \left( \frac{3}{4} + 4^{-2} \right) + 2$$. 2. **Recall the rules:** - Negative exponents mean reciprocal powers: $$a^{-n} = \frac{1}{a^n}$$. - Perform operations inside parentheses first. - Follow order of operations: parentheses, exponents, multiplication/division, addition/subtraction. 3. **Evaluate the exponent:** $$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$$. 4. **Simplify inside the parentheses:** $$\frac{3}{4} + \frac{1}{16} = \frac{12}{16} + \frac{1}{16} = \frac{13}{16}$$. 5. **Multiply by -2:** $$-2 \times \frac{13}{16} = -\frac{26}{16} = -\frac{13}{8}$$. 6. **Rewrite the expression:** $$\frac{8}{5} - 2 \left( \frac{3}{4} + 4^{-2} \right) + 2 = \frac{8}{5} - \frac{13}{8} + 2$$. 7. **Convert all terms to a common denominator to add:** - Common denominator for 5 and 8 is 40. - $$\frac{8}{5} = \frac{64}{40}$$. - $$-\frac{13}{8} = -\frac{65}{40}$$. - $$2 = \frac{80}{40}$$. 8. **Add all fractions:** $$\frac{64}{40} - \frac{65}{40} + \frac{80}{40} = \frac{64 - 65 + 80}{40} = \frac{79}{40}$$. **Final answer:** $$\frac{79}{40}$$ or 1.975.