`(2^2)^3 . 1/(2^5) . 4^3`
`=2^(2.3) . 1/(2^5) . (2^2)^3`
`= 2^6 . 1/(2^5) . 2^6`
`= 2^12 . 1/(2^5) = (2^12)/(2^5) = 2^7`
`(5^3)^4 . 1/(5^3) .125 :5^4`
`=` `5^7` `xx` `1` `:` `5^3` `xx` `5^3` `:` `5^4`
`=` `[(5^7) : (5^4)] .[(5^3) . (5^3)] . 1`
`=` `5^(7-4) . 5^(3+3) . 1`
`=` `5^3 . 5^6.1`
`=` `5^(3+6) . 1`
`=` `5^9` . 1`
`=` `5^9`
a) \(\left(2^2\right)^3.\dfrac{1}{2^5}.4^3\)
\(=\left(2^2\right)^3.\dfrac{1}{2^5}.\left(2^2\right)^3\)
\(=\left(2^2\right)^3.\left(2^2\right)^3.\dfrac{1}{2^5}\)
\(=\left(2^2\right)^6.\dfrac{1}{2^5}\)
\(=2^{12}.\dfrac{1}{2^5}\)
\(=\dfrac{2^{12}}{2^5}\)
\(=2^7\)
\(=128\)
b) \(\left(5^3\right)^4.\dfrac{1}{5^3}.125:5^4\)
\(=\left(5^3\right)^4.\dfrac{1}{5^3}.5^3:5^4\)
\(=\left(5^3\right)^4.\dfrac{5^3}{5^3}:5^4\)
\(=\left(5^4\right)^3:5^4\)
\(=\left(5^4\right)^2\)
\(=5^8\)