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a) \(4^{10}.8^{15}=\left(2^2\right)^{10}.\left(2^3\right)^{15}\)
\(=2^{20}.2^{45}\)
\(=2^{65}\)
\(a,4^{10}\cdot8^{15}=\left[2^2\right]^{10}\cdot\left[2\cdot2^2\right]^{15}=2^{20}\cdot2^{15}\cdot2^{30}=2^{20+15+30}=2^{65}\)
\(b,\frac{27^{16}}{3^{10}}=\frac{\left[3^3\right]^{16}}{3^{10}}=\frac{3^{48}}{3^{10}}=3^{48-10}=3^{38}\)
\(c,\frac{\left[2^9\cdot16+2^9\cdot34\right]}{2^{10}}=\frac{\left[2^9\left\{16+34\right\}\right]}{2^{10}}=\frac{2^9\cdot50}{2^{10}}=\frac{1}{2}\cdot50=25\)
\(d,\frac{3^4\cdot57-9^2\cdot21}{3^5}=\frac{3^4\cdot57-\left[3^2\right]^2\cdot21}{3^5}=\frac{3^4\cdot57-3^4\cdot21}{3^5}=\frac{3^4\left[57-21\right]}{3^5}=\frac{1}{3}\cdot36=12\)
a) $3^8:3^6=3^{8-6}=3^2$
$19^7:19^3=19^{7-3}=19^4$
$2^{10}:8^3=2^{10}:(2^3)^3=2^{10}:2^9=2^{10-9}=2^1$
$12^7:6^7=(12:6)^7=2^7$
$27^5:81^3=(3^3)^5:(3^4)^3=3^{15}:3^{12}=3^{15-12}=3^3$
b) $10^6:10=10^{6-1}=10^5$
$5^8:25^2=5^8:(5^2)^2=5^8:5^4=5^{8-4}=5^4$
$4^9:64^2=4^9:(4^3)^2=4^9:4^6=4^{9-6}=4^3$
$2^25:32^4=2^{25}:(2^5)^4=2^{25}:2^{20}=2^{25-20}=2^5$
$18^3:9^3=(18:9)^3=2^3$
\(\cdot3^8:3^6=3^{8-6}=3^2\)
\(\cdot19^7:19^3=19^{7-3}=19^4\)
\(\cdot2^{10}:8^3=2^{10}:\left(2^3\right)^3=2^{10}:2^9=2\)
\(\cdot12^7:6^7=\left(12:6\right)^7=2^7\)
\(\cdot27^5:81^3=\left(3^3\right)^5:\left(3^4\right)^3=3^{15}:3^{12}=3^3\)
\(\cdot10^6:10=10^{6-1}=10^5\)
\(\cdot5^8:25^2=5^8:\left(5^2\right)^2=5^8:5^4=5^4\)
\(\cdot4^9:64^2=4^9:\left(4^3\right)^2=4^9:4^6=4^3\)
\(2^{25}:32^4=2^{25}:\left(2^5\right)^4=2^{25}:2^{20}=2^5\)
\(18^3:9^3=\left(18:9\right)^3=2^3\)
\(\frac{2^{10}\cdot3^{10}-2^{10}\cdot3^9}{2^9\cdot3^{10}}=\frac{2^{10}\cdot3^9\left(3-1\right)}{2^9\cdot3^{10}}=\frac{2^{11}\cdot3^9}{2^9\cdot3^{10}}=\frac{2^2}{3}=\frac{4}{3}\)