If a=3^1/4 + 3^-1/4 and b=3^1/4 - 3^-1/4 than value of 3(a^2 + b^2) ^2 is 64.How i am getting 100/3

Please solved it in detail as i am getting answer but not the one given in text but as 100/3.

a = 3^(1/4) + 3^(-1/4) = (3^(2/4) + 1) / 3^(1/4) = (3^(1/2) + 1) / 3^(1/4)
b = 3^(1/4) - 3^(-1/4) = (3^(2/4) - 1) / 3^(1/4) = (3^(1/2) - 1) / 3^(1/4)

3 * (a^2 + b^2)^2 =>
3 * ((1/3^(1/4))^2 * (3^(1/2) + 1)^2 + (1/3^(1/4))^2 * (3^(1/2) - 1)^2)^2 =>
3 * (3^(-1/2) * ((3 + 2 * 3^(1/2) + 1 + 3 - 2 * 3^(1/2) + 1))^2 =>
3 * 3^(-1) * (3 + 3 + 1 + 1)^2 =>
1 * (8)^2 =>
1 * 64 =>
64

I don't know how you're getting 100/3, other than just simply making a mistake somewhere

a = 3^(1/4) + 3^(-1/4)
b = 3^(1/4) - 3^(-1/4)

a^2 =
(3^(1/4))^2 + 2 * 3^(1/4) * 3^(-1/4) + (3^(-1/4))^2 =
3^(1/2) + 2 * 1 + 3^(-1/2) =
3^(1/2) + 2 + 3^(-1/2)

b^2 =
(3^(1/4))^2 - 2 * 3^(1/4) * 3^(-1/4) + (3^(-1/4))^2 =
3^(1/2) - 2 * 1 + 3^(1/2) =
3^(1/2) + 3^(1/2) - 2

a^2 + b^2 =
3^(1/2) + 3^(-1/2) + 2 + 3^(1/2) + 3^(-1/2) - 2 =>
2 * 3^(1/2) + 2 * 3^(-1/2) =>
2 * (3^(1/2) + 3^(-1/2))

Square that:

2^2 * (3^(1/2) + 3^(-1/2))^2 =
4 * ((3^(1/2))^2 + 2 * 3^(1/2) * 3^(-1/2) + (3^(-1/2))^2) =
4 * (3 + 2 * 1/3) =
4 * (16/3) =
64/3

Multiply that all by 3

64

there are 2 ways

a^2 = (3^(1/2) + 3^(-1/2) + 2
b^2 = (3^(1/2) + 3^(-1/2) - 2

add to get a^2+b^2 = 2(3^(1/2) + 3^(-1/2))

square again to get (a^2+b^2)^2 = 4 (3+1/3+2) = 4(4/3) = 64/3

multiply by 3 to get the result

2)
add a and b to get a + b= 2 *3^(1/4)
subtract to get (a-b) = 2* 3^(-1/4)

2(a^2+b^2) = (a+b)^2 + (a-b)^2 = 4 * 3^(1/2) + 4 * 3^(-1/2)

or a^2+b^2 = 2 * (3^(1/2) + 2 * 3^(-1/2))

from here you can proceed as in (1)

Using brackets:-

a² = [3^(1/4) + 3^(-1/4)] [3^(1/4) + 3^(-1/4)]
a² = 3^(1/2) + 2 + 3^(-1/2)

b² = [3^(1/4) - 3^(-1/4) ] [3^(1/4) - 3^(-1/4)]
b² = 3^(1/2) - 2 + 3^(-1/2)

a² + b² = 2 [3^(1/2)] + 2 [3^(-1/2)]

a² + b² = 2[3^(-1/2)] [ 3 + 1 ]

a² + b² = 8 / 3^(1/2)

3 (a² + b²) = 8 [ 3^(1/2) ] = 8 √3

a^2 = 3^1/2 + 1/3^1/2 + 2 and b^2 = 3^1/2 + 1/3^1/2 - 2 (3^-1/4 is nothing but 1/3^1/4)

a^2 + b^2 = 2 x 3^1/2 + 2/3^1/2

(a^2 = b^2)^2 = 4 x 3 + 4/3 + 2 (4) i.e. 12 + 4/3 + 8

3(a^2 = b^2)^2 = 36 + 4 + 24 = 64.

a^2 = sq(3) + 2 + 1/sq(3)
b^2 = sq(3) - 2 + 1/sq(3)
a^2 + b^2 = 2sq(3) + 2/sq(3) = 2sq(3) + 2sq(3)/3 = (8/3)sq(3)
(a^2 + b^2)^2 = 64/9 * 3 = 64/3
3*(64/3)=64