Given expression
= (1 - 1/2) x (1 - 1/3) x (1 - 1/4) x (1 - 1/5) ...(1 - 1/m - 1) x (1 - 1/m)
= ((2 - 1)/2) x ((3 - 1)/3) ((4 - 1)/4) ((5 - 1)/5) ...((m-1 - 1)/m - 1) x ((m - 1)/m)
= (1/2) x (2/3) x (3/4) x (4/5) x ... x (m -2)/(m-1) x ((m - 1)/m
use the multiplication rule of Algebra,
= 1 x 1/m
= 1/m

Given equation is,
a + 1/a = ?3 .............................(i)
On squaring both sides
? a² + 1/a² + 2 = 3
? a² + 1/a² = 1 ..............................(ii)

Now, multiplying Equation (i) and (ii), we get
(a + 1/a) (a² + 1/a²) = ?3
? a³ + a/a² + a²/a + 1/a³ = ?3
? a³ + 1/a³ + ( a + 1/a ) = ?3
now put the value of a + 1/a in above equation,
? a³ + 1/a³ + ?3 = ?3 [from Eq. (i)]
? a³ + 1/a³ = 0
? ( a⁶ + 1 ) /a³ = 0
? a⁶ + 1 = 0 x a³
? a⁶ + 1 = 0
? a⁶ = -1
? a⁶ - 1/a⁶ + 2
put the value of a⁶
= (-1)⁶ - 1/(-1)⁶ + 2
= 1 - 1 + 2 = 2

(1 + 1/2) (1 + 1/3) (1 + 1/4) .... (1 + 1/150)
= 3/2 x 4/3 x 5/4 x ... 151/150
= (1/2) x 151
= 151/2 = 75.5

Given that, a + 1/b = 1
? a = (1 - 1/b) = (b - 1)/b
? 1/a = b/(b - 1) [reciprocal] ..........................(1)
and b + 1/c = 1
? 1/c = (1 - b)
? c = 1/(1 -b) .........................(2)
? c + 1/a ..........................(3)
put the value of c from equ. (2) and 1/a from equ. (1) in equ. (3)
? c + 1/a = 1/(1 -b) + b/(b -1)
? c + 1/a = 1/(1 -b) - b/(1 - b)
? c + 1/a = (1 - b)/(1 - b)
? c + 1/a = 1

(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

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

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

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

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 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