1. **State the problem:** Simplify the expression $$\frac{13 + (-3)^2 + 4(-3) + 1 - [-10 - (-6)]}{\left[4 + 5\right] \div \left[4^2 - 3^2(4 - 3) - 8\right] + 12}$$ using BEDMAS (Brackets, Exponents, Division/Multiplication, Addition/Subtraction). 2. **Simplify the numerator step-by-step:** - Calculate the exponent: $(-3)^2 = 9$ - Calculate the multiplication: $4(-3) = -12$ - Simplify inside the brackets: $-10 - (-6) = -10 + 6 = -4$ So numerator becomes: $$13 + 9 - 12 + 1 - [-4] = 13 + 9 - 12 + 1 + 4$$ 3. **Add and subtract in numerator:** $$13 + 9 = 22$$ $$22 - 12 = 10$$ $$10 + 1 = 11$$ $$11 + 4 = 15$$ Numerator = $15$ 4. **Simplify the denominator step-by-step:** - Simplify inside the first bracket: $4 + 5 = 9$ - Simplify inside the second bracket: - Calculate exponents: $4^2 = 16$, $3^2 = 9$ - Calculate inside parentheses: $4 - 3 = 1$ - Multiply: $9 \times 1 = 9$ - Calculate inside second bracket: $16 - 9 - 8 = 16 - 17 = -1$ 5. **Divide inside denominator:** $$9 \div (-1) = -9$$ 6. **Add 12 to the division result:** $$-9 + 12 = 3$$ Denominator = $3$ 7. **Form the simplified fraction:** $$\frac{15}{3}$$ 8. **Simplify the fraction by dividing numerator and denominator by 3:** $$\frac{\cancel{3}5}{\cancel{3}1} = 5$$ **Final answer:** $$5$$