If a = ( ?2 -1)^1/3, then the value of (a - a^-1)³ + 3 ( a - a ^-1 ) is?

Given equation is,
?28 - 6?3 = ?3a + b
By expanding equation,
? 1 + 27 - 6?3 = ?3a + b
? ?(1)² + (3?3)² - 2 x 3?3 = ? ?3a + b
? ?(1 - 3?3)² = ?3a + b
? 1 - 3?3 = ?3a + b
On comparing the left and right side equation, we get,
a = -3, b = 1
? a + b = -3 + 1 = - 2

Given, x + a/x = 1
? x² + a = x .................(i)
? x² - x = -a ................(ii)
Now, ( x² + x + a ) / ( x³ - x²)
( x² + a + x ) / ( x³ - x²)
= ( x + x )/ x(x² - x) [from Eq. (i)]
= 2x/x(-a) [from Eq. (ii) ]
= 2/-a
= -2/a

Given expression can be written as
= 1/(1 x 4) + 1/(4 x 7) + 1/(7 x 10) + 1/(10 x 13) + 1/(13 x 16)
= 1/3{ 3/(1 x 4) + 3/(4 x 7) + 3/(7 x 10) + 3/(10 x 13) + 3/(13 x 16) }
= 1/3{ (4 - 1)/(1 x 4) + (7 - 4)/(4 x 7) + (10 - 7 )/(7 x 10) + (13 - 10)/(10 x 13) + (16 - 13/(13 x 16) }
= 1/3[(1 - 1/4) + (1/4 - 1/7) + (1/7 - 1/10) + (1/10 - 1/13) + (1/13 - 1/16)]
= 1/3[1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 - 1/13 + 1/13 - 1/16]
= 1/3[1 - 1/16]
= 1/3[ (16 - 1)/16]
= 1/3[ (15)/16]
= 1 x 15/ 3 x 16
= 5/16

Given that,
x = ( ?2 + 1 ) / ( ?2 - 1 )
Expanding given equation by multiplying and dividing with ?2 + 1
we will get,
? x = ( ( ?2 + 1 ) / ( ?2 - 1 ) ) x ( ( ?2 + 1 ) / ( ?2 + 1 ) )
? x = ( ?2 + 1 ) x ( ?2 + 1 ) / ( ?2 - 1 ) x ( ?2 + 1 )
? x = ( ?2 + 1 ) ² / ( ?2 - 1 ) x ( ?2 + 1 ) [ Use the formula (A + B)(A - B) = A² - B² ]
? x = ( ?2 + 1 ) ² / ( ?2² - 1² ) [ Use the formula (A + B)² = A² + 2AB + B² ]
? x = ( 2 + 1 + 2?2 ) / (2 - 1)
? x = 3 + 2?2 .....................(i)
and given in question x - y = 4 ?2
? y = x - 4?2
Put the value of x [ from Equation (i) ]
? y = 3 + 2?2 - 4?2
? y = 3 - 2?2
Now, x² + y² = (3 + 2?2)² + (3 - 2?2)
Use the Algebra formula and solve the equation.
[ Use the formula (A + B)² = A² + 2AB + B² ]
[ Use the formula (A - B)² = A² - 2AB + B² ]

? x² + y² = 9 + 8 + 12?2 + 9 + 8 - 12?2
? x² + y² = 34

Let a = 8.73 and b = 4.27
Then, given expression =( a³ + b³ ) / ( a² - ab + b² )
As we Know the Algebra formula,
a³ + b³ = (a + b) (a² - ab + b²)
Apply the above algebra formula in given above expression,
= (a + b) (a² - ab + b²) / ( a² - ab + b² )
= (a + b)
= 8.73 + 4.27 = 13

Given, x = ( ?3 + 1) / ( ?3 - 1 )
y = ( ?3 - 1 ) / ( ?3 + 1 )
Then , x + y = { ( ?3 + 1) / ( ?3 - 1 ) } + { ( ?3 - 1 ) / ( ?3 + 1 ) }
? x + y = { (?3 + 1)² + (?3 - 1)² } / (?3 - 1) (?3 + 1)
? x + y = ( 3 + 1 +2?3 + 3 + 1 - 2?3 ) / ( 3 - 1 )
? x + y = 8/2
? x + y = 4 --------------------(1)
Now xy = ( ?3 - 1 / ?3 + 1 ) x ( ?3 + 1 / ?3 - 1 )
xy = (3 - 1) / (3 - 1) = 2/2 = 1------------------(2)
Given in question,
? x²/y + y²/x
= ( x³ + y³ )/xy ------------------- (3)
Apply the algebra formula, A³ + B³ = (A + B)³ - 3AB(A + B)
= (x + y)³ - 3xy(x + y)/xy
Put the value of x + y from (1) and xy from (2) in Equation (3), we will get
= (4)³ - 3 x 1 x (4)/1
= 64 - 3 x 1 x 4
= 64 - 12
= 52

Given in question,
x + a/x = b
? ( x² + a )/x = b
? x² + a = bx .........................(i)
? bx - x² = a .........................(ii)
Now Given in question,
( x² + bx + a ) / ( bx² - x³ )
? ( x² + a + bx ) / ( bx² - x³ )
Put the value of x² + a from equation (i) in above equation,
? ( bx + bx )/(bx² - x³)
= (2bx)/( bx² - x³ )
= (2bx)/x( bx - x² )
= 2b/(bx - x²)
Put the value of bx - x² from equation (ii) in above equation, we will get
= 2b/a

51.300 + 7.078 + 1.380 + 0.900 = 60.658

For the product of 3.5413 x 2.1
Consider 35413 x 21 = 743673
Total decimal places = 4 + 1
? 3.5413 x 2.1=7.43673

Multiplying 3.242 x 14=4.5388
Now divide 4.5388 by 100
So, 4.5388 ÷ 100= 0.045388
? Shift the decimal two places to the left as 100