Since a * b = a + b + a / b
? 12 * 4 = 12 + 4 + 12 / 4
= 12 + 4 + 3 = 19
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
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,
(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)
= 2.247 + 1.730 x 1.023) - (2.247 x 1.023)
= (2.247 + 1.730 + 1.023) [from Eq.(i)]
= 5.000 = 5
? = 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
(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 ?
? 430 + 24 - 3 = 43 x 7
And comparing the exponents both the sides
? 451 = 43 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
Given that, x + 1/x = 2 ...................(i)
On squaring both sides, we get
(x + 1/x)2 = 4
? x2 + 1/x2 + 2 = 4
? x2 + 1/x2 = 2 ..................................(ii)
Now, we have
(x - 1/x)2 = (x2 + 1/x2) - 2x X 1/x
now put the value of x2 + 1/x2
(x - 1/x)2 = 2 - 2 = 0 [from Eq. (ii)]
? x - 1/x = 0
x + 1/x = 6
On squaring both sides, we get
(x + 1/x)2 = (6)2
? x2 + 1/x2 + 2 = 36
? x2 + 1/x2 = 34
On squaring both sides, we get
(x2 + 1/x2)2 = (34)2
? x4 + 1/x4 + 2 = 1156
? x4 + 1/x4 = 1154
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