Đáp án và giải thích các bước giải:
`a)`
Thay `x=25` , ta được :
`A={\sqrt[25]+2}/{\sqrt[25]-1}`
`A={5+2}/{5-1}`
`A=7/4`
`b)`
`DKXD:x≥0;x\ne1`
`B={\sqrt[x]-2}/{\sqrt[x]+3}+{\sqrt[x]-3}/{\sqrt[x]+2}-{x-6\sqrt[x]-22}/{x+5\sqrt[x]+6}`
`B={(\sqrt[x]-2)(\sqrt[x]+2)+(\sqrt[x]+3)(\sqrt[x]-3)-x+6\sqrt[x]+22}/{(\sqrt[x]+2)(\sqrt[x]+3)}`
`B={x+6\sqrt[x]+9}/{(\sqrt[x]+2)(\sqrt[x]+3)}`
`B={(\sqrt[x]+3)^2}/{(\sqrt[x]+2)(\sqrt[x]+3)}`
`B={\sqrt[x]+3}/{\sqrt[x]+2}`
`c)`
`P=A.B=1+4/{\sqrt[x]-1}`
`P∈`$\mathbb{Z}$ `⇔` `1+4/{\sqrt[x]-1}∈`$\mathbb{Z}$
Mà : `x∈`$\mathbb{Z}$
`⇒` `\sqrt[x]∈`$\mathbb{Z}$
`⇒` `\sqrt[x]-1∈`$\mathbb{Z}$
`⇒` `4\vdots\sqrt[x]-1`
`⇒` `\sqrt[x]-1∈Ư(4)={±1,±2,±4}`
`⇔` `\sqrt[x]∈{0,2,-1,3,-3,5}`
Mà : `x≥0`
`⇒` `x∈{0,4,9,25}`
