$\text{52.}$
$\text{a,}$ `45^{10}.5^{20}/75^{15}`
$\text{=}$ `(3^2.5)^{10}.5^{20}/(3.5^2)^{15}`
$\text{=}$ `3^{20}.5^{10}.5^{20}/3^{15}.5^{30}`
$\text{=}$ `3^{20}.5^{30}/3^{15}.5^{30}`
$\text{=}$ `3^{5}.1/1.1`
$\text{=}$ `3^{5}`
$\text{=}$ `243`
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$\text{b,}$ `(0,8)^{5}/(0,4)^{6}`
$\text{=}$ `(4/5)^2` $\text{:}$ `(2/5)^6`
$\text{=}$ `(2^2/5)^2` $\text{.}$ `(5/2)^6`
$\text{=}$ `2^10/5^5` $\text{.}$ `5^6/2^6`
$\text{=}$ `2^10.5^6/5^5.2^6`
$\text{=}$ `2^4.5/1.1`
$\text{=}$ `2^4.5`
$\text{=}$ `16.5`
$\text{=}$ `80`
---------------------------------------------
$\text{c,}$ `2^15.9^4/6^6.8^3`
$\text{=}$ `2^15.3^8/(3.2)^6.2^9`
$\text{=}$ `2^15.3^8/3^6.2^6.2^9`
$\text{=}$ `2^15.3^8/3^6.2^15`
$\text{=}$ `1.3^2/1.1`
$\text{=}$ `1.3^2`
$\text{=}$ `1.9`
$\text{=}$ `9`
--------------------------------------
$\text{53.}$
$\text{1 = }$ `3^0`
$\text{243 = }$ `3^5`
`1/3` $\text{=}$ `3^(-1)`
`1/3` $\text{=}$ `3^(-2)`
