If x=1/7+4√3 and y=1/7-4√3, then find the value of (i)x2+y2 (ii)x3+y3
Well x^2=1/49 + 48+8/7 *sqrt3
and y^2=1/49 - 48+8/7 *sqrt3
check those. Then add them.
To find the value of x^2 + y^2, we can start by calculating x^2 and y^2 separately.
(i) Calculating x^2:
x^2 = (1/7 + 4√3)^2
To simplify this, we can use the binomial theorem expansion:
x^2 = (1/7)^2 + 2*(1/7)*(4√3) + (4√3)^2
x^2 = 1/49 + 8/√3 + 48
(ii) Calculating y^2:
y^2 = (1/7 - 4√3)^2
Using the same process as above:
y^2 = (1/7)^2 - 2*(1/7)*(4√3) + (4√3)^2
y^2 = 1/49 - 8/√3 + 48
Now, we can add x^2 and y^2 together to find the value:
x^2 + y^2 = (1/49 + 8/√3 + 48) + (1/49 - 8/√3 + 48)
Simplifying the expression:
x^2 + y^2 = 2/49 + 96
(i) To find the value of x^3 + y^3, we can use the formula for the sum of cubes:
x^3 + y^3 = (x + y)(x^2 - xy + y^2)
First, let's calculate (x + y):
x + y = (1/7 + 4√3) + (1/7 - 4√3)
Simplifying the expression:
x + y = 2/7
Next, we need to calculate (x^2 - xy + y^2):
(x^2 - xy + y^2) = [(1/7 + 4√3)^2 - (1/7 + 4√3)(1/7 - 4√3) + (1/7 - 4√3)^2]
Expanding and simplifying:
(x^2 - xy + y^2) = (1/49 + 8/√3 + 48) - (1 - 48) + (1/49 - 8/√3 + 48)
(x^2 - xy + y^2) = 96 + 2/49
Now, we can find the value of x^3 + y^3 by multiplying (x + y) with (x^2 - xy + y^2):
x^3 + y^3 = (2/7) * (96 + 2/49)
Multiplying both terms:
x^3 + y^3 = 192/7 + 4/49