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

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

Half of the cloth = 37.5/2
From 1 meter he will make 8 pieces
=> in 37.5/2 ---- ?

37.5/2 x 8

= 150.