1. **State the problem:** Simplify the expression $$\frac{3^{8} \cdot 2^{7} \cdot 4^{2} \cdot 25^{2}}{3^{6} \cdot 2^{5} \cdot 2^{4} \cdot 5^{4}} - \frac{243^{\frac{1}{2}}}{9^{\frac{1}{4}}} - \frac{1}{3^{-3}}$$. 2. **Recall exponent rules:** - $a^{m} \cdot a^{n} = a^{m+n}$ - $\frac{a^{m}}{a^{n}} = a^{m-n}$ - $(a^{m})^{n} = a^{m \cdot n}$ - $a^{0} = 1$ 3. **Simplify the first big fraction:** - Rewrite bases where possible: $4 = 2^{2}$, $25 = 5^{2}$. - So numerator: $3^{8} \cdot 2^{7} \cdot (2^{2})^{2} \cdot (5^{2})^{2} = 3^{8} \cdot 2^{7} \cdot 2^{4} \cdot 5^{4} = 3^{8} \cdot 2^{11} \cdot 5^{4}$. - Denominator: $3^{6} \cdot 2^{5} \cdot 2^{4} \cdot 5^{4} = 3^{6} \cdot 2^{9} \cdot 5^{4}$. - Divide powers with same base: $$\frac{3^{8}}{3^{6}} = 3^{8-6} = 3^{2}$$ $$\frac{2^{11}}{2^{9}} = 2^{11-9} = 2^{2}$$ $$\frac{5^{4}}{5^{4}} = 5^{0} = 1$$ - So the first fraction simplifies to $3^{2} \cdot 2^{2} = 9 \cdot 4 = 36$. 4. **Simplify the second term:** $\frac{243^{\frac{1}{2}}}{9^{\frac{1}{4}}}$ - Express bases as powers of 3: $243 = 3^{5}$, $9 = 3^{2}$. - So numerator: $243^{\frac{1}{2}} = (3^{5})^{\frac{1}{2}} = 3^{\frac{5}{2}}$. - Denominator: $9^{\frac{1}{4}} = (3^{2})^{\frac{1}{4}} = 3^{\frac{2}{4}} = 3^{\frac{1}{2}}$. - Divide powers: $3^{\frac{5}{2}} / 3^{\frac{1}{2}} = 3^{\frac{5}{2} - \frac{1}{2}} = 3^{2} = 9$. 5. **Simplify the third term:** $\frac{1}{3^{-3}} = 3^{3} = 27$. 6. **Combine all terms:** $$36 - 9 - 27 = 0$$ **Final answer:** $0$