Question

what should be subtracted from a^4+b^4 to make it perfect square​

Answer 1

Step-by-step explanation:

Question:

a⁴+b⁴

Solution:

a^4 + b^4 - 2 a^2 b^2 = (a^2 - b^2)^2

a^4 + b^4 - b^4 = a^4 = (a^2)^2

a^4 + b^4 - a^4 = b^4 = (b^2)^2

(a-b)^4 = a^4 - 4a^3 * b + 6 a^2 b^2 - 4 a*b^3 + b^4 = ((a-b)^2)^2

→ a^4 + b^4 - (4a^3 * b - 6 a^2 b^2 + 4 a*b^3) = ((a-b)^2)^2

So, you can substract either:

2*a^2 * b^2

or

b^4

or

a^4

or

4 a^3 b - 6 a^2 b^2 + 4 a b^3

Example:

Let a=3; b=2

a^4 = 81

b^4 = 16

a^4 + b^4 = 97

2 a^2 b^2 = 2 * 9 * 4 = 72

97 - 72 = 25 = (a^2 - b^2)^2 = (9–4)^2 = 5^2

97 - 16 = 81 = 9^2

97 - 81 = 16 = 4^2

4 a^3 b - 6 a^2 b^2 + 4 a b^3 = 4*27*2 - 6*9*4 + 4*3*8 = 216 - 216 + 96

97 - 96 = 1 = 1^2 = ((a-b)^2)^2