1)\[\frac{6^{4}\times 8^{2}\times 27^{2}}{2^{6}\times 3^{6}\times 2^{4}\times 3^{3}}=?\]
Explanation: \[\frac{\left ( 2\times 3 \right )^{4}\times \left ( 2^{3} \right )^{2}\times \left ( 3^{3} \right )^{2}}{2^{6}\times 3^{6}\times 2^{4}\times 3^{3}}\] \[\frac{2^{4}\times 3^{4}\times 2^{6}\times 3^{6} }{2^{6}\times 3^{6}\times 2^{4}\times 3^{3}}\] \[=\frac{2^{10}\times 3^{10}}{2^{10}\times 3^{9}}=3^{1}=3\]
2) \[\left ( -5 \right )^{3}\times \left ( -2 \right )^{4}\times \left ( -1 \right )^{5}\]
Explanation:-125 × 16 × -1 = 2000
3) \[\left ( \frac{x^{a}}{x^{b}} \right )^{(a^{2}+ab+b^{2})}\: \times \left ( \frac{x^{a}}{x^{b}} \right )^{\left ( b^{2+bc+c^{2}} \right )}\: \times \left ( \frac{x^{c}}{x^{a}} \right )^{\left ( c^{2}+ca+a^{2} \right )}\]
Explanation:\[x^{\left ( a+-b \right )\left ( a^{2+ab+b^{2}} \right )}\: \times x^{\left ( b-c \right )\left ( b^{2}+ca+a^{2} \right )}\]\[x^{\left ( c-a \right )\left ( c^{2+ca+a^{2}} \right )}\]\[x^{\left ( a^{3}-b^{3} \right )}\times x^{\left ( b^{3}-c^{3} \right )}\times x^{\left ( c^{3}-a^{3} \right )}\]\[x^{\left ( a^{3}-b^{3} +b^{3}-c^{3}+c^{3}-a^{3}\right )}=x^{0}=1\]
4)\[\frac{\left ( 2^{n}+2^{n-1} \right )}{2^{n+1}-2^{n}}=?\]
Explanation:\[\frac{\left ( 2^{n-1}\left ( 2+1 \right ) \right )}{2^{n}\left ( 2-1 \right )}=\frac{3}{2}\]