(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] =?

? = 5 - [3/4 + {5/2 - (1/2 + 1/6 - 1/7)}]/2
= 5 - [3/4 + {5/2 - (1/2 + 7 - 6/42 )}]/2
= 5 - [3/4 + {5/2 - (1/2 + 1/42)}]/2
= 5 - [3/4 + {5/2 - (21 + 1/42)}]/2
= 5 - [3/4 + {5/2 - 22/42)}]/2
= 5 - [3/4 + {105 - 22/42)}]/2
= 5 - [3/4 + 83/42]/2 = 5 - [63 + 166/84]/2
= 5 - 229/84/2 = 420 - 229/84/2 = 191/84/2
= 191/84 x 2 = 191/168 = 1²³/₁₆₈

Let population of cities A and B become equal after x year.
According to the question.
Population of city A = Population of city B
Then,
136000 - 2400 x = 84000 + 1600 x.
? 4000x = 52000
? x = 52/4 = 13 year

Let total number of apples in the crate = a
Then, number of bruised apples = a/40 .
Number of unsaleable apples
since 4 bruise apples containing 3 unsaleable apples.
therefore 1 bruise apples will contain 3/4 unsaleable apples
hence a/40 bruise apples will contain 3/4 x a/40 unsaleable apples
= 3/4 x a/40 = 3a/160
According to the question.
3a/160 = 9
? a = 9 x 160/3
a = 3 x 160 = 480

Let the worker remains absent for x days.
Then, his working days = (120 - x) days.
According to the question,
20 x Working day - 3 x Absent day = 560
20(120 - x) - 3x = 560
? 2400 - 20x - 3x = 560
? x = 2400 - 560/23 = 1840/23
= 80 days

Solution 1 :
Let the number of ? 50 notes = x, then the number of ? 100 notes = (170 - x)
According to question.
50x + 100(170 - x) = 10000
divide by 50 the above equation, we will get
? x + 2(170 - x) = 200
? x + 340 - 2x = 200
? -x = 200 - 340 = -140
? x = 140
? Required amount ? 50 x 140
= ? 7000


Solution 2 :
Let the number of ? 50 notes = x and the number of ? 100 notes = y.
according to given question,
x + y = 170 .............(1)
50x + 100y = 10000............(2)

Divide the equation (2) by 10 , we will get
5x + 17y = 1000............(3)
Multiply the equation (1) by 5
5x + 5y = 850 ------------(4)
Subtract the (4) equation from (3) equation, we will get,
5x +10y - ( 5x + 5y ) = 1000 - 850
? 5x +10y - 5x - 5y = 1000 - 850
? 10y - 5y = 1000 - 850
? 5y = 150
? y = 150/5
? y = 30
Number of ? 100 notes = y = 30;
since x + Y = 170
therefore x = 170 - y
? x = 170 - 30
? x = 140
Number of ? 50 notes = x = 140

a² + 1 = a ? a + 1/a = 1
On squaring both sides, we get
a² + 1/a² + 2 = 1
? a² + 1/a² = -1
On cubing both sides, we get
(a² + 1/a²)³ = (-1)³
? a⁶ + 1/a⁶ + 3a² x 1/a²(a² + 1/a²) = -1
? a⁶ + 1/a⁶ + 3 x (-1) = -1
Now, a⁶ + 1/a⁶ + 1 = 3
As, a¹² + a⁶ + 1 can also be written as a⁶ + 1/a⁶ + 1
? a¹² + a⁶ + 1 = 3

Since a * b = a + b + a / b
? 12 * 4 = 12 + 4 + 12 / 4
= 12 + 4 + 3 = 19

(4 x 4 x 4 x 4 x 4 x 4)⁵ x (4 x 4 x 4)⁸ ÷ (4)³ = (64)^?
Apply the law of Fractional Exponents and Laws of Exponents
if a multiply n times a x a x a x....up to n times, then
a x a x a x a ......up to n times = aⁿ
By simplifying the equation
? (4⁶)⁵ x (4³)⁸ x 1/(4)³ = (4³)^?
? (4)³⁰ x (4)²⁴/(4)³ = (4)^{3 x ?}
? 4^{30 + 24 - 3} = 4^{3 x 7}
And comparing the exponents both the sides
? 4⁵¹ = 4^{3 x ?} ? 3 x ? = 51
? ? = 51/3 = 17

Given equation are
3x + 2y = 12 ..............(i)
xy = 6 .........................(ii)
On squaring Eq. (i) on both sides, we will get
(3x + 2y)² = (12)²
? 9x² + 4y² + 12xy = 144
put the value of xy
? 9x²+ 4x² = 144 - 72 = 72
? 9x²+ 4x² = 72

x + 1/x = 3 ........................ (i)
On squaring both side, we will get
(x + 1/x)² = (3)²
Use the Square algebra formula (a + b)² = a² + 2ab + b²
? x² + 1/x² + 2 = 9
? x² + 1/x² = 7 ..................(ii)
Again squaring both sides, we will get
(x² + 1/x²)² = (7)²
Use the Square algebra formula (a + b)² = a² + 2ab + b²
? x⁴ + 1/⁴ + 2 = 49
? x⁴ + 1/x⁴ = 47 .....................(iii)
On cubing the equation (i) both side, we will get
(x + 1/x)³ = (3)³
Use the cube algebra formula (a + b)³ = a³ + 3a²b + 3ab² + b³
? x³+ 1/x³ + 3(x + 1/x) = 27
? x³ + 1/x³ + 9 = 27 [? (x + 1/x) = 3]
? x³ + 1/x³ = 18 ...................(iv)
On multiplying Eqs. (i) and (iii) , we get
? (x⁴ + 1/x⁴) (x + 1/x) = 47 x 3
? x⁵ + 1/x⁵ + x³ + 1/x³ = 141
? x⁵ + 1/x⁵ + 18 = 141 [from Eq. (iv)]
? x⁵ + 1/x⁵ = 123