If x = ( ?2 + 1 ) / ( ?2 - 1 ) and x - y = 4?2, then the value of (x² + y²) is?

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

We know that, a³ + b³ + c³ - 3abc = (a + b + c ) ( a² + b² + c² - ab - bc - ca )
? (a + b + c) = ( a³ + b³ + c³ - 3abc ) / ( a² + b² + c² - ab - bc - ca ) .......................(i)
Given that in question,
(2.247)³ + (1.730)³ + (1.023)³ - 3 x 2.247 x 1.730 x 1.023 / (2.247)² + (1.730)² + (1.023)² - (2.247 x 1.730) - (1.730 x 1.023) - (2.247 x 1.023)
Apply the formula given in Equation (i) and we will get,
= ( a + b + c )
= ( 2.247 + 1.730 + 1.023) [ from eq. (i) ]
= 5.000 = 5

The given question is,
(70 ÷ 100)^{? + 3} = (0.49)⁴ x (0.343)⁴ ÷ (0.2401)⁴
As we know that
70 ÷ 100 = 0.7
0.49 = 0.7 ²
0.343 = 0.7 ³
0.2401 = 0.7 ⁴
Put these value in given question , we will get
? (0.7)^{? + 3} = (0.7²)⁴ x ( 0.7³ )⁴ ÷ ( 0.7⁴ )⁴
? (0.7) ^{? + 3} = 0.7 ^{2 x 4} x 0.7 ^{3 x 4} ÷ 0.7 ^{4 x 4}
? (0.7) ^{? + 3} = 0.7 ⁸ x 0.7 ¹² ÷ 0.7 ¹⁶
Apply the law of Fractional Exponents and Laws of Exponents
(a^m)(aⁿ) = a^m+n
a^m÷aⁿ=a^m-n
Or
a^m/aⁿ=a^{m - n}
we will get,
? (0.7)^{? + 3} = (0.7) ^{8 + 12 - 16}
? (0.7)^{? + 3} = (0.7) ^{20 - 16}
? (0.7)^{? + 3} = (0.7) ⁴
On comparing the exponents both sides, we get
? + 3 = 4
? ? = 4 - 3 = 1

Simplify 2³ to 8
= 5/8 x 8/5÷4/9
By using simple algebraic rule: a ÷ b/c = a x c/b
= 5/8 x 8/5 x 9/4
= (5×8×9)/(?8×5×4)
= (5×9)/(?5×4)
= (9)/(?4)
= 9/4

(a² + 1/a²) = 17/4
subtracts 2 and Add 2 from above equation, we will get
? a² + 1/a² - 2 + 2 = 17/4
? a² + 1/a² - 2a x 1/a + 2 = 17/4
Now apply the formula, x² + x² - 2xy
? (a - 1/a)² + 2 = 17/4
? (a - 1/a)² = 17/4 - 2
? (a - 1/a)² = (17 - 8)/4
? (a - 1/a)² = 9/4
? (a - 1/a) = ?9/4
(a -1/a) = 3/2
After that, cubing On both sides, we get
(a -1/a)³ = (3/2)³
Apply the formula (x ? y)³ = x³ ? 3x²y + 3xy² ? c³
? (x ? y)³ = x³ ? 3xy(x - y) ? y³
? a³ - 1/a³ - 3 x a x 1/a (a - 1/a) = 27/8
? a³ - 1/a³ = 27/8 + 3 x (3/2)
? a³ - 1/a³ = 27/8 + 9/2
? (a³ - 1/a³) = 63/8

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, 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 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 that,
a = (?2 - 1)^1/3
By cubing the both side of the given equation.
a³ = ?2 - 1
Then 1/a³ = 1/?2 - 1
Now multiply and divide the above equation by ?2 + 1
? 1/a³ = ( ?2 + 1 ) / ( ?2 + 1 ) x 1/(?2 - 1 )
? 1/a³ = ( ?2 + 1 ) x 1 / ( ?2 + 1 ) x (?2 - 1 ) [ Use the formula (A + B)(A - B) = A² - B² ]
? 1/a³ = ( ?2 + 1 ) / ( 2 - 1 )
? 1/a³ = ?2 + 1
According to the question
( a - a^-1 )³ + 3 ( a - a^-1 )
= (a - 1/a)³ + 3(a - 1/a) Use the formula [ ( A - B)³ = A³ - B³ + 3AB(A - B) ]
= a³ - 1/a³ - 3 a x 1/a (a -1/a) + 3(a - 1/a)
= a³ - 1/a³ - 3 (a -1/a) + 3(a - 1/a)
= a³ - 1/a³
Pu the value of a³ and 1/a³
= ?2 -1 - ( ?2 + 1 )
= ?2 -1 - ?2 - 1
= -2

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