Here, 1-xm-n-1+1-xn-m-1 = 1-xmxn-1+1-xnxm-1 = xn-xmxn-1+xm-xnxm-1 = xnxn-xm-xmxn-xm = .....

LHS = xbxcbc × xcxaca × xaxbab xb-cbc = × xc-aca × xa-bab = xb-cbc × xc-aca × xa-bab = xb-cbc+c-aca+a-bab = xab-c+ bc-a+c(a-b)abc = xab-ac+bc-ab+ac-bcabc = x0abc = x0 = 1 = RHS Hence, .....

Here, =ax-yx+y-z× ay-zy+z-x× az-xz+x-y = ax2+ xy-xz-xy- y2 + yz× ay2+ yz-xy-zy- z2 + zx× az2+zx-zy-xz- x2 + xy = ax2+ xy-xz-xy- y2 + yz+y2+ yz-xy-zy- z2 + zx +z2+zx-zy-xz- x2 + xy = a0 = .....

Here, 2x×3-2x2x+2-2x+1 = 2x×3-2x2x.4-2x.2 = .....

Here,     ∛(9x-2y7 ) × ∛(3x5 y-1 ) = (9x-2y7×3x5 y-1)1/3 = (27x-2+5y7-1)1/3 = (27x3 y6 )1/3 = (33 x3 y6 )1/3 = (3xy2 )3×1/3 =  3xy2 . .....

Here, a-5334-45 = a-5334-45 = a-53×34×-45 =a .....

LHS = (xb/xc )(1/bc)  ⨉(xc/xa )(1/ca)  ⨉ (xa/xb )(1/ab)          = (xb-c)1/bc  ⨉ (xc-a)1/ca   ⨉ (xa-b)1/ab          = x((b-c)/bc)  ⨉x((c-a)/ca)  ⨉ .....

Here,         3x+3 + 3x+1 = 119  Or, 3x.33 + 3x.31 = 109  Or, 3x (33 + 31) = 109  Or, 3x (27+3) = 109  Or, 3x ×30 = 109  Or, 3x = 109⨉30  Or, 3x = 127  Or, 3x = 133  Or, 3x = .....

Here,      xa2+b2x-aba-b× xb2+c2x-bcb-c× xc2+a2x-cac-a = xa2+ab+b2(a-b)×xb2+bc+c2(b-c)×xc2+ca+a2(c-a) = xa3-b3+b3-c3+ c3-a3 = x0 = 1 .....

LHS = 11+ xa-b + xa-c + 11+ xb-c + xb-a+ 11+ xc-a + xc-b =  1xaxa + xaxb +xaxc + 1xbxb + xbxc +xbxa + 1xcxc + xcxa +xcxb =  1xa(1xa + 1xb +1xc ) + 1xb(1xb + 1xc +1xa ) + 1xc(1xc + 1xa +1xb ) =  1xa(xb .xc + xa .xc+ xa .xb xa .....

Here, 64-13 = 1643 = 182×123 = ∛( 1/8) = (1/2)3 . 1/3 = 1/2.  .....

Here,        21-x+ 2x-1 = 52 Or, 21. 2-x + 2x . 2-1 = 52 Or, 22x + 2x 2 = 52 Let 2x = a Then, 2a + a2 = 52 Or, 4+ a22a= 52 Or, 4+ a2a = 5 Or, 4 + a2 = 5a Or, a2  - 5a + 4 = 0 Or, a2  - (4 + 1)a + 4 = 0 Or, a2  - .....

Here, LHS = (xl/xm )(l+m) ⨉ (xm/xn )(m+n) ⨉ (xn/xl )(n+l)            = (x l-m)l+m ⨉ (xm-n)m+n ⨉  (xn-l)n+l          = x(l2-m2) ⨉ x(m2-n2 ) ⨉ x(n2-l2 )    .....

Here,     (49x + 72x-1) / (8 ×72x-1)  = (72x+72x×7-1) / (8×72x×7-1 ) = {72x (1+7-1)} / (8×72x×7-1 ) = (1+1/7)/(8×1/7) = ((7+1)/7)/(8/7) = (8/7)/(8/7) = 8/7  × 7/8 = .....

Here,      3x +13x = 103 Or, 3x.3x+ 13x = 103 Or, 32x+ 13x = 103 Or, 3x2+13x = 103 Let 3x = a, Then, a2+1a = 103 Or, 3 (a2 + 1) = 10a Or, 3a2 + 3 = 10a Or, 3a2 – 10a + 3 = 0 Or, 3a2 – (9+1)a + 3 = 0 Or, 3a2 – 9a-a + .....

Here, 11n+2-55.11n-111n.116 = 11n.121-55.11n1111n.116 = 11n121-511n.116 = .....

Here,      4x + 4-x = 16 Or, 4x + 14x = 25716 Or, 4x.4x+ 14x = 25716 Or, 42x+ 14x = 25716 Or, 4x2+14x = 25716 Let 4x = a, Then, a2+1a = 25716 Or, 16 (a2 + 1) = 257a Or, 16a2 + 16 = 257a Or, 16a2 – 257a + 16 = .....

Here,               LHS = al2am2 l+m   ×am2an2m+n   ×an2al2n+l   = al2-m2l+m ×am2-n2m+n ×an2-l2n+l = al2-m2l+m ×am2-n2m+n ×an2-l2n+l = al+m(l-m)l+m .....

Here, a+b-73.a+b13 = a+b-73 . a+b13 = a+b-73+13 = a+b-2 = .....

Here,    3√2/(3- √(6 )) - 2√3/(3+ √(6 ))  = (3√2)/(3- √(6 )) ⨉ (3+ √(6 ))/(3+ √(6 )) - (2√3)/(3+ √(6 )) ⨉ (3- √(6 ))/(3- √(6 )) = (3√2 .....

Here, 1/(1+ √(2 )) + 1/(√2+ √(3 )) + 1/(√3+ √(4 )) 1st term:     1/(1+ √(2 ))  = 1/(1+ √(2 )) ⨉ (1- √(2 ))/(1- √(2 )) = (1- √(2 ))/((1)2- (√2)2 ) = (1- √(2 .....

Here,        √(x+7)+√x = 28÷√(x+7) Or,  √(x+7) (√(x+7) + √x) = 28 Or, (√(x+7))2 + √(x(x+7)) = 28 Or, (x + 7) + √(x2+ 7x) = 28 Or, √(x2+ 7x) = 28 – x .....

Here,         √(x+5) - √x     = 1 Or, √(x+5) = 1 + √x  Squaring both sides, we get       x + 5 = 12 + 2. 1. √x + √x 2 Or, x + 5 = 1 + 2√x + x Or, 5 .....

Here,        (√x  + √a)/(√x- √a) = 4 or, √x  + √a = 4(√x- √a) or, √x  + √a = 4√x- 4√a or, 5√a = 3√x Squaring both .....

Here,    (√x+ √a)÷(√x- √a) =  (√x+ √a)÷(√x- √a) ⨉ (√x+ √a)÷(√x+ √a) =  (√x  + √a)2 ÷ (√x2-√a2 .....

Here, 1÷(x+ √(x2-1)) + 1÷(x- √(x2-1)) Taking LCM, = [x- √(x2-1)+x+ √(x2-1)]÷[(x+ √(x2-1)) (1/(x- √(x2-1)] = 2x÷ [(x2 - √(x2-1)2 ] = 2x ÷ [x2 - .....

Here,      √(x+65)-√x = 45÷√(x+65) Or,  √(x+65) (√(x+65) - √x) = 45 Or, (√(x+65))2 - √(x(x+65)) = 45 Or, (x + 65) - √(x2+ 65x) = 45 Or, √(x2+ 65x) = x + .....

Here,      (3√x-4)÷(√x+ 2) = (15+3√x  )÷(√x+ 40). Or, .....

Here,        √(2x+25) - √(2x-15) = 4 Or,  √(2x+25)  = 4 + √(2x-15)  Squaring on both sides,   √(2x+25)2 = (4 + √(2x-15) )2 Or, 2x + 25  = 42 + 2.4. √(2x-15) + .....

Here,       √(x+9) + √x = 45÷√(x+9) Or, √(x+9) (√(x+9) + √x)  = 45 Or, (√(x+9))2 + √(x(x+9)) = 45 Or, (x + 9) + √(x2+ 9x)    = 45 Or, x + 9 + √(x2+ 9x) .....

Here,       √(x+24) - √x = 2 Or, √(x+24)  = 2 + √x Squaring on both sides, (√(x+24))2  = (2 + √x)2 Or, x + 24   = 22 + 2.2. √x + √x2 Or, x + 24   = 4 + 4√x + .....

Here,      ∜(32x7 y11 ) ÷ ∜(2x3 y7 ) =  ∜(32x7 y11 ) ÷∜(2x3 y7 ) =  ∜(32x7 y11)÷(2x3 y7 ) =  ∜(24x4 y4 ) =  2xy. .....

Here, (√6-2)÷[5√2- √(72 )- 2√48+3√27] = (√6-2)÷[5√2- √(36 ⨉ 2 )- 2√(16 ⨉ 3)+3√(9 ⨉ 3)] = (√6-2)÷[5√2- 6√(2 )- 2⨉ 4√3+3⨉3√3] = .....

Here, (a+b)a-ba+b = a+b2a-ba+b = a+b2a-ba+b = a+ba-b = a2-b2  Which is required .....

Here,     8√(1÷32) - 6√(1÷2) + 3√(1÷2).   = 8√(1÷(16 ⨉ 2)) - 6√(1÷2) + 3√(1÷2) = 8÷4 ⨉ 1÷√2 - 6 ÷√2 + .....

Here,     ∛(375p7 q) ÷ ∛(3pq-8) = ∛{(375p7 q)/∛(3pq-8)} = ∛{(375p7 q)/3pq-8)} = ∛(125p6q9)  =  5p2q3 .  .....

Here,    ∛(54x8 y4 ) ÷∛(2x5 y) = ∛(54x8 y4 )/∛(2x5 y) = ∛{(54x8 y4)/(2x5 y)} = ∛(27x3 y3 ) = 3xy .....

Here,   (3∛128+2∛16-2∛250  )÷∛54 = (3∛(64 ⨉ 2)+2∛(8 ⨉ 2)-2∛(125 ⨉ 2)  )÷∛(27 ⨉ 2) = (3 ⨉ 4∛2+2 ⨉ 2∛2-2 ⨉ 5∛2 )÷ 3∛2 = (12∛2+4∛2-10∛2 )÷3∛2 = (16∛2-10∛2 .....

Here,     √(x+2) = 3 Squaring both sides, (√(x+2) )2= 32 Or, x + 2  = 9 Or, x        = 9-2 Or, x        = 7 ∴    x        = 7 When x= .....

Here       (x-3)÷(√x+ 1) = 2(√x- 1)÷3 Or, 3(x-3) = 2 (√x- 1) (√x+1) Or, 3x -9 = 2 (√x2 - 1) Or, 3x – 9 = 2 (x - 1) Or, 3x – 9 = 2x – 2 Or, 3x – 2x = 9 - 2 Or, x = .....

Here,    4∛375 - 7∛192 + 2∛3000 = 4∛(125 ⨉ 3) - 7∛(64 ⨉ 3) + 2∛(1000 ⨉ 3) = 4 ⨉ 5 ∛3 – 7 ⨉ 4∛3 + 2 ⨉ 10∛3 = 20 ∛3 – 28∛3 + 20∛3 = 40 ∛3 – 28∛3 = 12∛3,  .....

We have,   √(x+13)=5, Squaring both sides,  Or, x + 13 = 52 Or, x =25-13 ∴     x=12, .....

Here,     6∛250 ÷5∛54 = (6∛250)÷(5∛54) = 6÷5 ⨉ ∛(250÷54) = 6÷5 ⨉ ∛(125÷27) = 6÷5 ⨉ 5÷3 = 2 .....

We have,    2√x = 3, Squaring both sides, or, x= 32 or, x = 9 Thus, x=9 , .....

Here,    12÷√72 =12÷√(2×6⨉6) =12÷(6×√2) =2÷√2 =√2×√2÷√2 =√2 , .....

Here,    2√3 (√12  + √27) = 2√36 + 2√81 = 2 ⨉ 6 + 2 ⨉ 9 = 12 + 18 = 30 .....

Here,   ∛(4x2 y) . ∛(4xy5 ) = ∛(4x2 y ⨉ 4xy5 )    = ∛(16x3 y6 ) = ∛(8 ⨉ 2 ⨉ x3 y6 ) = 2xy2∛2 =2∛2  xy2 , .....

Here,       (x-1)÷ (√x- 1) = 3 Or, (√x-1)(√x+1)÷ (√x- 1) = 3 Or, √x+1= 3 Or, √x = 2 Squaring both sides,we get     x = 4. Hence, the solution is x = .....

Here,     √(x2- 4)÷√(x+2) = √[(x2- 4)÷(x+2)] = √[(x+2)(x-2)} .....

Here,       [√(1+ x2 )+ √(1-x2 )]÷ [√(1+ x2 )- √(1-x2 )] = 2 Or, √(1-x2 )+ 2√(1-x2 )= 2√(1+ x2 )-√(1+ x2 ) Or, 3√(1- x2) = √(1+ x2 ) Squaring on both sides,   .....