1.

(a) `sqrt15xxsqrt35.` (b) `2sqrt3 div3sqrt27.` (c ) Muliple `root(3)(3)" by " root(4)(2).`(d) Divide `""root(6)(5)" by "root(3)(10).`

Answer» `(a) (sqrt15)(sqrt35)=sqrt((15)(35))=sqrt((5)(3)(5)(7))=5sqrt(21).`
(b) `2sqrt3div3sqrt27=(2sqrt3)/(3sqrt27`
`=(2sqrt3)/((3)sqrt(3^(2)(3)))=(2sqrt3)/((3)(3)sqrt3)9/3.`
(c ) `""^(3)sqrt3=3^(1//3)and""^(4)sqrt2=2^(1//4)`
The LCM of 3 and 4 is 12
`therefore3^(1//3)=3^(4//12)=""^(12)sqrt(3^(4))`
`2^(1//4)=2^(3//12)=""^(12)sqrt(2^(3))`
`(""^(3)sqrt3)(""^(4)sqrt2)=(""^(12)sqrt(3^(4)))(""^(12)sqrt(2^(3)))`
`=""^(12)sqrt((3^(4))(""^(12)sqrt(2^(3))))`
`=""^(12)sqrt((81)(8))=""^(12)sqrt(648).`
(d) `""^(6)sqrt5=5^(1//6)`
LCM of 3 and 6 is 6
`""^(3)sqrt10=10^(1//3)=10^(2//6)=""^(6)sqrt(10^(2))=""^(6)sqrt100`
`therefore(""^(6)sqrt5)/(""^(3)sqrt10)=(""^(6)sqrt10)/(""^(6)sqrt100)`
`=""^(6)sqrt((5)/(100))=""^(6)sqrt((1)/(20)).`


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