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\(8^2.32^4=2^{2.3}.2^{5.4}=2^6.2^{20}=2^{26}\)
\(64^3.256^2=2^{6.3}.2^{8.2}=2^{18}.2^{16}=2^{34}\)
\(27^3.9^4.243=3^{3.3}.3^{2.4}.3^5\)
\(3^9.3^8.3^5=3^{22}\)
\(25^3.125^2=5^{2.3}.5^{3.2}=5^6.5^6=5^{12}\)
\(\left(3^8.3^{15}\right):\left(3^7.3^{14}\right)\)
=\(3^{23}:3^{21}\)
=\(3^2\)
\(\left(4^{16}.4^{12}\right).\left(4^{20}:2\right)\)
=\(4^{28}.\left(2^{40}:2\right)\)
\(4^{28}.2^{39}=2^{56}.2^{39}=2^{95}\)
a, \(\left(\dfrac{1}{2}+1\right).\left(\dfrac{1}{3}+1\right).\left(\dfrac{1}{4}+1\right)...\left(\dfrac{1}{999}+1\right)\)
\(=\dfrac{3}{2}.\dfrac{4}{3}.\dfrac{5}{4}...\dfrac{1000}{999}\)
\(=\dfrac{3.4.5...1000}{2.3.4...999}\)
\(=\dfrac{1000}{2}\)\(=500\)
b, \(\left(\dfrac{1}{2}-1\right).\left(\dfrac{1}{3}-1\right).\left(\dfrac{1}{4}-1\right)...\left(\dfrac{1}{1000}-1\right)\)
\(=\dfrac{-1}{2}.\dfrac{-2}{3}.\dfrac{-3}{4}...\dfrac{-999}{1000}\)
\(=\dfrac{\left(-1\right).\left(-2\right).\left(-3\right)...\left(-999\right)}{2.3.4...1000}\)
\(=\dfrac{-1}{1000}\)
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