1. **State the problem:** Simplify the expression $$P = \frac{9 \cdot 40 \cdot 2^{x+3} \cdot 41 + 3^{y+4} - 7 \cdot 40 \cdot 2^{x+2} \cdot 41}{5 \cdot 40 \cdot 3^{y+2} \cdot 41 - 3 \cdot 40 \cdot 2^{x+4} \cdot 41 + 3^{y+3} + 2^{x}}$$ given that $$2^{x} = 3^{y}$$. 2. **Rewrite the expression using the given equality:** Since $$2^{x} = 3^{y} = k$$, substitute to simplify powers: - $$2^{x+3} = 2^{x} \cdot 2^{3} = k \cdot 8 = 8k$$ - $$2^{x+2} = k \cdot 4 = 4k$$ - $$2^{x+4} = k \cdot 16 = 16k$$ - $$3^{y+4} = 3^{y} \cdot 3^{4} = k \cdot 81 = 81k$$ - $$3^{y+3} = k \cdot 27 = 27k$$ - $$3^{y+2} = k \cdot 9 = 9k$$ 3. **Substitute into numerator:** $$9 \cdot 40 \cdot 8k \cdot 41 + 81k - 7 \cdot 40 \cdot 4k \cdot 41$$ Calculate constants: - $$9 \cdot 40 \cdot 8 \cdot 41 = 9 \cdot 40 \cdot 328 = 9 \cdot 13120 = 118080$$ - $$7 \cdot 40 \cdot 4 \cdot 41 = 7 \cdot 40 \cdot 164 = 7 \cdot 6560 = 45920$$ So numerator: $$118080k + 81k - 45920k = (118080 + 81 - 45920)k = 72141k$$ 4. **Substitute into denominator:** $$5 \cdot 40 \cdot 9k \cdot 41 - 3 \cdot 40 \cdot 16k \cdot 41 + 27k + k$$ Calculate constants: - $$5 \cdot 40 \cdot 9 \cdot 41 = 5 \cdot 40 \cdot 369 = 5 \cdot 14760 = 73800$$ - $$3 \cdot 40 \cdot 16 \cdot 41 = 3 \cdot 40 \cdot 656 = 3 \cdot 26240 = 78720$$ So denominator: $$73800k - 78720k + 27k + k = (73800 - 78720 + 27 + 1)k = (-4892)k$$ 5. **Simplify the fraction:** $$P = \frac{72141k}{-4892k}$$ Cancel common factor $$k$$: $$P = \frac{\cancel{k} \cdot 72141}{-4892 \cdot \cancel{k}} = \frac{72141}{-4892} = -\frac{72141}{4892}$$ 6. **Final answer:** $$P = -\frac{72141}{4892}$$ This fraction can be left as is or approximated numerically if needed.