1. **Stating the problem:** Calculate the value of the expression for problem 279: $$279\quad 2 \cdot \left[\frac{(4^2 \cdot 4^3)^2}{4^{10}}\right] - 7 \cdot 8^0 + \frac{9^2}{3^2}$$ 2. **Recall the rules and formulas:** - Powers with the same base multiply by adding exponents: $a^m \cdot a^n = a^{m+n}$ - Powers raised to powers multiply exponents: $(a^m)^n = a^{m \cdot n}$ - Division of powers with the same base subtracts exponents: $\frac{a^m}{a^n} = a^{m-n}$ - Any number to the zero power is 1: $a^0 = 1$ 3. **Calculate inside the brackets:** - Calculate $4^2 \cdot 4^3 = 4^{2+3} = 4^5$ - Then square it: $(4^5)^2 = 4^{5 \cdot 2} = 4^{10}$ - Divide by $4^{10}$: $\frac{4^{10}}{4^{10}} = 4^{10-10} = 4^0 = 1$ 4. **Substitute back:** $$2 \cdot 1 - 7 \cdot 8^0 + \frac{9^2}{3^2}$$ 5. **Calculate powers:** - $8^0 = 1$ - $9^2 = 81$ - $3^2 = 9$ 6. **Simplify the fraction:** $$\frac{81}{9} = 9$$ 7. **Calculate the entire expression:** $$2 \cdot 1 - 7 \cdot 1 + 9 = 2 - 7 + 9$$ 8. **Final sum:** $$2 - 7 + 9 = (2 - 7) + 9 = -5 + 9 = 4$$ **Answer:** $4$