1. **State the problem:** Simplify the expression $$y = \frac{(-5)^2 - 4^2 + \left( \frac{1}{5} \right)^0}{3^{-2} + 1}$$. 2. **Recall important rules:** - Any number raised to the power 0 is 1, i.e., $$a^0 = 1$$ for $$a \neq 0$$. - Negative exponents mean reciprocal, i.e., $$a^{-n} = \frac{1}{a^n}$$. - Order of operations: exponents first, then multiplication/division, then addition/subtraction. 3. **Calculate numerator:** - $$(-5)^2 = (-5) \times (-5) = 25$$ - $$4^2 = 16$$ - $$\left( \frac{1}{5} \right)^0 = 1$$ So numerator is $$25 - 16 + 1$$. 4. **Simplify numerator:** $$25 - 16 + 1 = 9 + 1 = 10$$ 5. **Calculate denominator:** - $$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$ So denominator is $$\frac{1}{9} + 1$$. 6. **Simplify denominator:** $$\frac{1}{9} + 1 = \frac{1}{9} + \frac{9}{9} = \frac{10}{9}$$ 7. **Write the fraction:** $$y = \frac{10}{\frac{10}{9}}$$ 8. **Divide by a fraction:** $$y = 10 \times \frac{9}{10}$$ 9. **Cancel common factors:** $$y = \cancel{10} \times \frac{9}{\cancel{10}} = 9$$ **Final answer:** $$y = 9$$