Here,         ∛(81a4 b7 )   = ∛(3×3×3×3×a×a×a×a×b×b×b×b×b×b×b)                   .....

Here,     (3√2 - 2√3)2 = (3√2)2 – 2.3√2. 2√3 + (2√3)2 = 9 ⨉ 2 – 12√6 + 4 ⨉ 3  = 18 - 12√6 + 12 = 30-12√6, .....

Here,   √(9-x2 ) ÷ √(3-x) - √(x2- 9) ÷ √(x-3)  =√[{(3+x )(3-x)} ÷ (3-x)] - √[{(x+3)(x-3)}÷(x-3)] =√(3+x) – √(x+3)  =0, .....

Here,       √(x+7) - √(2x-14) = √(x-5) Or, √(x+7) - √(x-5) = √(2x-14) . Squaring both sides,       (√(x+7) - √(x-5) )2= √(2x-14)2 Or, (√(x+7) )2- .....

Here,      2 (x +2)          = 1 + √(4x2+ 10x+25) Or, 2x + 4       = 1+ √(4x2+ 10x+25) Or, 2x + 4 – 1 = √(4x2+ 10x+25) Or, 2x + 3       = √(4x2+ .....

Here,      (3x-4)÷(√3x+ 2)   = 2 + (√3x- 2)

Here,    (√(x+y)+ √(x-y))÷(√(x+y )- √(x-y)) = (√(x+y)+ √(x-y))÷(√(x+y )- √(x-y)) ⨉ (√(x+y)+ √(x-y))÷(√(x+y )+ √(x-y)) = .....

Here,    7∜625 - 13∛216 + 3√289 = 7 ⨉ 5 – 13 ⨉ 6 + 3 ⨉ 17 = 35 – 78 + 51 = 86 – 78 = 8.  .....

Here, 73-5248-18 = 73-523×4⨉4-2×3⨉3 = 73-5243-32 ⨉ 43+3243+32 28×3+216-206-15×248-18 = .....

Here, The LCM of the roots of the term 2, 3, 4 is 12. Now, 32 = 31/2 =312×66 = 3612 = 3612 = 12√729 53= 51/3 = 513×44 = 5412 = .....

Here, 5+35-3+ 5-35+3 = 5+32+5-325-35+3 = 5+215+3+5-215+352-32 =  162 = 8, .....

Here,     7 163+254 3+3128 3-4250 3 = 78×2 3+2 27×23+364×2 3-42⨉125 3 = 723×2 3+233×2 3+343×2 3-42⨉53 3 = .....

Here, 2-1x74.32x-34 = 2-1x7.25.x-34 = 2-1+5x7-34 = 24x44 = 2x, .....

Here, 45+245-500+625 = 3 ⨉3 ×5+5×7⨉7  -5×10⨉10+25×25 = 35+75-105+25 = 25, which is required .....

Here,     √18   =√(2×3×3)   =3√2 . .....

Solution: x2÷(1- √(1-x2  )) = x2÷(1- √(1-x2  )) ⨉ (1+ √(1-x2 ))÷(1+ √(1-x2  )) = (x2  (1+ √(1-x2  )))÷((1- √(1-x2  ))(1+ √(1-x2  )) = [x2 .....

Here,    √(45a2 ) - √(80a2 ) + 6a√5 ÷ (5√(5a2 ) = (√(9 ⨉ 5 ⨉ a2 )- √(16⨉ 5 ⨉ a2 ) + 6a√5)÷(5a√5) =(3a√5- 4a√5+ .....

Here,     1÷∛(2x-3) = 2 Taking cube root on both sides, (1÷∛(2x-3))3 = 23 Or, 1÷(2x-3)  = 8 Or, 1             = 8(2x - 3) Or, 1             = .....

Here,    3√27  + 2√12 - 2√3 = 3√(9 ⨉ 3)  + 2√(4 ⨉ 3) - 2√3 = 3 ⨉ 3√3  + 2 ⨉ 2 √3 - 2√3 = 9√3  + 4√3 - 2√3 = 13√3  - .....

Here,   (√54- √96+ √150)÷√24 =(√(9 ⨉ 6)- √(16 ⨉ 6)+ √(25 ⨉ 6))÷√(4 ⨉ 6) =(3√6- 4√6+ 5√6)÷(2√6) =(8√6- .....

The rationalizing factor of ∛x+∛y = (∛(x2 )+∛(y2 ) .....

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 .....

Here,    (2∛250-∛128)/(3∛16) =(2∛(125 ⨉2)-∛(64 ⨉2))÷(3∛(8⨉2)) =(2 ⨉ 5∛2-4∛2)÷(3⨉2∛2) =(10∛2-4∛2)÷(6∛2) =6∛2÷(6∛2) =1 , .....

Here,      √[(a2- 9) ÷ (a - 3)] =  √[(a2- 32) ÷ (a-3) ] = √[(a-3)(a+3) ÷ (a-3) ] = √(a+3),  .....

Here,      (a+b)  ÷ √(a2- b2 ) =  √(a+b)2 ÷√(a2- b2 ) = √[(a+b)2÷(a2- b2 )]  = √[(a+b)2÷(a+b)(a-b)] =  √[(a+b)÷(a-b) ],  .....

Here, 5x-45x+2 = 4 - 5x-32 or, 5x+25x-25x+2 = 8- 5x+32 or, 2 5x-2 = 11 - 5x or, 25x – 4 = 11- 5x or, 35x = 15 or, 5x = 5 or, 5x = 25. Therefore, x = .....

Here, The LCM of the roots of the term 4, 3, 6 is 12. Now, x24 = x2/4 = x24×33=x612=x612 x43 = x4/3 = x43×44=x1612=x1612 x3 6 = x3/6 = .....

Here,    (2∛192-4∛(81)  + 3∛(24))÷ (∛648- ∛375) = (2∛(64 ⨉ 3) - 4∛(27⨉ 3)  + 3∛(8 ⨉ 3 )) ÷ (∛(216 ⨉ 3) - ∛(125⨉ 3)) = (2 ⨉ 4∛3 - 4 ⨉ 3∛(3)  + 3⨉2 ∛(3))÷(6∛3 .....

Here,    7∜625 - 13∛216 + 3√289 = 7 ⨉ 5 – 13 ⨉ 6 + 3 ⨉17 = 35 – 78 + 51 = 86 – 78 = 8 ,  .....

Here,       (x-1)÷(√x+ 1) = 4 + (√x-1)÷2 Or,  (√x2-12)÷(√x+ 1) = 4 + (√x-1)÷2 Or,  (√x-1)(√x+ 1)÷(√x+ 1) = 4 + .....

Here,      √(2x+1) + x = 7 Or, √(2x+1) = 7-x. Squaring both sides,       (√(2x+1) )2 = (7-x)2 Or, 2x + 1 = 72 - 2.7.x + x2 Or, 2x + 1 = 49 - 14x + x2 Or, x2 - 14x + 49 – 2x – 1 = 0 Or, x2 .....

Here,    (2√3 - 3√2)2 = (2√3)2 – 2. 2√3. 3√2 + (3√2)2 = 4 ⨉ 3 – 12 ⨉ √6 + 9 .....

Here,     a√(ab3 ) ÷ b√(a3 b) = (a√(ab3 )÷(b√(a3 b) = a÷b √{(ab3÷(a3 b)} = a÷b √(b2÷a2 ) = a÷b ⨉ b÷a = 1 , .....

Here,     (√75+√27  )÷√3 .....

Here,     ∜80 + ∜405 = ∜(16 ⨉ 5) + ∜(81 ⨉ 5) =  2∜5 + 3∜5 =  5∜5, .....

Here,    (4√72- 3√50)÷√18 =[4√(36 ⨉2)- 3√(25 ⨉ 2)]÷√(9 ⨉ 2) =[4 ⨉ 6√2- 3 ⨉ 5√2]÷3√2 =[24√2- 15√2]÷3√2 =9√2 .....

Here,    √27  + 5√12 - 3√48 = √(9 ⨉ 3)  + 5√(4 ⨉ 3) - 3√(16 ⨉ 3) = 3√3  + 5 ⨉ 2√3 – 3 ⨉ 4√3 = 3√3  + 10√3 – 12√3 = √3( .....

Here,        3√99 - 5√44 + 4√11 =  3√(9 ⨉ 11) - 5√(4 ⨉ 11) + 4√11 =  3 ⨉ 3√11 – 5 ⨉ 2√11 + 4√11 =  9√11 - 10√11 + .....

Here,     (2√3 - 3√2)2 = (2√3)2 – 2. 2√3. 3√2 + (3√2)2 = 4⨉ 3 – 12 ⨉ √6 + 9 ⨉ 2 = 12 - 12√6 + 18 = 30 - 12√6 .....

Here,         3√99 - 5√44 + 4√11 =  3√(9 ⨉ 11) - 5√(4 ⨉ 11) + 4√11. = 3 ⨉ 3√11 – 5 ⨉ 2√11 + 4√11 = 9√11 - 10√11 + 4√11 = 13√11 .....

Here, 128a34÷2-1a-94 = 128a342-1a-94 = 27a32-1a-94 = 28a124 = (22a3)44 = .....

Here, 27a33 .25a2b2 4 = 33a33 . 52a2b24 = 3a.(5ab)2/4 =3a√(5ab). .....

Here,    ∜(16x12 y16 ) ÷∛(8x6 y9 ) = ∜(16x12 y16 )÷∛(8x6 y9 ) = (2x3 y4)÷(2x2 y3 ) = xy.       .....

Here,    (√48+√12)÷√3 = (√(16  ⨉ 3)+√(4  ⨉ 3))÷√3 = (4√3+2√3)÷√3 = 6√3÷√3 = .....

Here,     4√2+5√2+7√2   =(4+5+7)√2   =16√2 , .....

Here,

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Here, 1st expression = a2 + ab + b2 . 2nd expression = a4 + a2b2 + b4                          = (a2)2 + (b2)2 + a2b2                .....

Here, 1st expression = x2 + 3x                         = x(x + 3). 2nd expression = x3 + 27                          .....

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