Simplify the following (i) `sqrt45-3sqrt20+4sqrt5` (ii) `sqrt(24)/8 + sqrt54/9` (iii) `root4(12) xx root7(6)` (iv) `4sqrt28 div 3sqrt7 div root3(7)` (v)` 3sqrt3+2sqrt27 + 7/(sqrt3)` (vi) `(sqrt3-sqrt2)^(2)` (vii) `root4(81)-8root3(216)+15root5(32)+ sqrt225` (viii) `3/sqrt8+ 1 / sqrt2` (ix) `(2sqrt3)/3- (sqrt3)/6`

Explanation

`sqrt45-3sqrt20+4sqrt5= sqrt(3xx3xx5)-3sqrt(2xx2xx2)+4sqrt5`
` 3sqrt5-3xx2sqrt5+4sqrt5`
`3sqrt5-6sqrt5+4sqrt5=sqrt5`
`sqrt24/8+sqrt54/9=sqrt(2xx2xx2xx2xx3)/8+sqrt(3xx3xx3xx2)/9`
`(2sqrt6)/8 + (3sqrt6)/9=sqrt6/4+sqrt6/3`
`(3sqrt6+4sqrt6)/12=(7sqrt6)/12`
`root4(12)xxroot7(6)=(12)^(1//4)xx(6)^(1//7)`
`(2xx2xx3)^(1//4)xx(2xx3)^(1//7)=2^(1//4).2^(1//4).3^(1//4).2^(1//7).3^(1//7)`
`2^(1/4+1/4+1/7)xx3^(1/4+1/7)` `2^(9//14)xx3^(11//28)=root14(2^(9))root28(3^(11)) " " [ x^(m).x^(n)=x^(m+n)]`
`root28(2^(18))root28(33^(11))=root28(2^(18)xx3^(11)) " " [ root na= root(mn)(a^(m))`
(iv) `4sqrt28+3sqrt7 div root3(7) =(4sqrt(4xx7)div3sqrt7)divroot3(7)`
` ((8sqrt7)/(3sqrt7))div(7)^(1/3)`
`8/3div7^(1/3)= 8/(3root3(7))`
(v) `3sqrt3+2sqrt27+7/sqrt3=3sqrt3+2sqrt(3xx3xx3)+7/sqrt3xxsqrt3/sqrt3`
`3sqrt3+ 6sqrt3+(7sqrt3)/3=9sqrt3+(7sqrt3)/3`
`3sqrt3+6sqrt3+(7sqrt3)/3=9sqrt3+(7sqrt3)/3`
`(27sqrt3+7sqrt3)/3=(34sqrt3)/3`
(vi) `(sqrt3-sqrt2)^(2)+(sqrt2)^(2)-2sqrt3xxsqrt2 ` [ using identity , `(a-b)^(2)= a^(2+b^(2)-2ab]`
`3+2-2sqrt(3xx2)=5-2sqrt6`
(vii) `root4(81)-8root3(216)+15root5(32)+sqrt225`
`(81)^(1/4)-8xx(6^(3))^(1/3)15xx(32)^(1/5)+sqrt((15)^(2))`
` (3^(4))^(1/4)-8xx(6^(3))^(1/3)+15xx2^(5xx(1/5))+15`
`3^(1)-8xx6^(1)+15xx2^(1)+15`
3-48+30+15
48-48=0
`3/sqrt8+1/sqrt2=3/(sqrt(2xx2xx2))+1/sqrt2=3/(2sqrt2)+1/sqrt2`
` (3+2)/(2sqrt2)+1/sqrt2`
`(5sqrt2)/(2xx2)=5/(2sqrt2)xxsqrt2/sqrt2`
`(5sqrt2)/(2xx2)=(5sqrt2)/4`
(ix) `(2sqrt3)/3-sqrt3/6=(4sqrt3-sqrt3)/6=(3sqrt3)/6=sqrt3/2`