Given in question,
x + y = 18
By using algebraic formula
? x2 + y2 = (x + y)2 - 2xy
Put the value of x + y and xy as per given question,
? x2 + y2 = (18)2 - 2 x 72
? x2 + y2 = 324 - 144
? x2 + y2 = 180
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
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 = 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 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
(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
We know that algebraic formula,
(x + y)3 = x3 + y3 + 3xy (x + y)
put the value of x + y in given equation. [ given, x + y = 1]
1 = x3 + y3 + 3xy X 1
? x3 + y3 + 3xy = 1
p/q + q/p = (p2 + q2)/pq
Apply the formula of algebra
a 2+ b2 = (a + b)2 - 2ab
p/q + q/p = ( (p + q)2 - 2pq ) / pq
By substituting the pq and p + q values in given equation.
p/q + q/p = ( (p + q)2 - 2pq ) / pq
p/q + q/p = ((10)2 - 2 x 5 ) / 5
p/q + q/p = (100 - 10 )/ 5 = 90/5 = 18
p/q + q/p = 90/5 = 18
p/q + q/p = 18
Let, (m - n) = a,
(n -r) = b
(r - m) = c,
Now a + b + c = (m - n) + (n -r) + (r - m)
? a + b + c = m - n + n - r + r - m
? a + b + c = 0............................. (i)
As we know the Algebra formula,
a3 + b3 + c3 ? 3abc = (a+b+c) X 1/2[(a?b)2+(b?c)2+(a?c)2]
Put the value of a + b + c from equation (i).
? a3 + b3 + c3 ? 3abc = 0 X 1/2[(a?b)2+(b?c)2+(a?c)2]
? a3 + b3 + c3 ? 3abc = 0
? a3 + b3 + c3 = 3abc
? Given expression in question is
[ (m - n)3 + (n - r)3 + (r - m)3 ]/ 6(m - n) (n - r) (r - m)
= ( a3 + b3 + c3 ) / 6abc
= 3abc/6abc
= 1/2
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
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