use the binomial theorem to expand and simplify (y^2-2)^6 )(y^2+2)^6
( a - b ) * ( a + b ) = a ^ 2 - b ^ 2
( y ^ 2 - 2 ) * ( y ^ 2 + 2 ) =
( y ^ 2 ) ^ 2 - ( 2 ^ 2 ) ^ 2 =
y ^ 4 - 4
( y ^ 2 - 2 ) ^ 6 * ( y ^ 2 + 2 ) ^ 6 =
[ ( y ^ 2 - 2 ) * ( y ^ 2 + 2 ) ] ^ 6 =
( y ^ 4 - 4 ) ^ 6
( a - b ) ^ 6 =
a^6-6 a^5 b+15 a^4 b^2-20 a^3 b^3+15 a^2 b^4-6 a b^5+b^6
( y ^ 4 - 4 ) ^ 6 =
( y ^ 4 ) ^ 6 - 6 * ( y ^ 4 ) ^ 5 * 4 + 15 * ( y ^ 4 ) ^ 4 * 4 ^ 2 -
20 * ( y ^ 4 ) ^ 3 * 4 ^ 3 + 15 *( y ^ 4 ) ^ 2 * 4 ^ 4 - 6 * ( y ^ 4 ) * 4 ^ 5 + 4 ^ 6 =
y ^ 24 - 6 * y ^ 20 * 4 + 15 * y ^ 16 * 16 - 20 y ^ 12 * 64 + 15 y ^ 8 * 256
- 6 * y ^ 4 * 1024 + 4096 =
y ^ 24 - 24 y ^ 20 + 240 y ^ 16 - 1280 y ^ 12 + 3840 y ^ 8 - 6144 y ^ 4 + 4096
( y ^ 2 - 2 ) ^ 6 * ( y ^ 2 + 2 ) ^ 6 =
( y ^ 4 - 4 ) ^ 6 =
y ^ 24 - 24 y ^ 20 + 240 y ^ 16 - 1280 y ^ 12 + 3840 y ^ 8 - 6144 y ^ 4 + 4096
To expand and simplify the expression (y^2 - 2)^6 (y^2 + 2)^6 using the binomial theorem, we can follow these steps:
1. Recognize that both expressions have the form (a - b)^n and (a + b)^n, where a = y^2 and b = 2.
2. Apply the binomial theorem for each expression:
For (y^2 - 2)^6:
(y^2 - 2)^6 = C(6, 0)(y^2)^6(-2)^0 + C(6, 1)(y^2)^5(-2)^1 + C(6, 2)(y^2)^4(-2)^2 + C(6, 3)(y^2)^3(-2)^3 + C(6, 4)(y^2)^2(-2)^4 + C(6, 5)(y^2)^1(-2)^5 + C(6, 6)(y^2)^0(-2)^6
For (y^2 + 2)^6:
(y^2 + 2)^6 = C(6, 0)(y^2)^6(2)^0 + C(6, 1)(y^2)^5(2)^1 + C(6, 2)(y^2)^4(2)^2 + C(6, 3)(y^2)^3(2)^3 + C(6, 4)(y^2)^2(2)^4 + C(6, 5)(y^2)^1(2)^5 + C(6, 6)(y^2)^0(2)^6
3. Simplify each term in both expanded expressions:
For (y^2 - 2)^6:
C(6, 0)(y^2)^6(-2)^0 = 1(y^2)^6(1) = y^12
C(6, 1)(y^2)^5(-2)^1 = 6(y^2)^5(-2) = -12y^10
C(6, 2)(y^2)^4(-2)^2 = 15(y^2)^4(4) = 60y^8
C(6, 3)(y^2)^3(-2)^3 = 20(y^2)^3(-8) = -160y^6
C(6, 4)(y^2)^2(-2)^4 = 15(y^2)^2(16) = 240y^4
C(6, 5)(y^2)^1(-2)^5 = 6(y^2)^1(-32) = -192y^2
C(6, 6)(y^2)^0(-2)^6 = 1(y^2)^0(64) = 64
Therefore, (y^2 - 2)^6 simplifies to y^12 - 12y^10 + 60y^8 - 160y^6 + 240y^4 - 192y^2 + 64.
For (y^2 + 2)^6, perform the same steps as above but with positive coefficients.
4. Multiply the simplified expressions:
(y^2 - 2)^6 (y^2 + 2)^6 = (y^12 - 12y^10 + 60y^8 - 160y^6 + 240y^4 - 192y^2 + 64) * (y^12 + 12y^10 + 60y^8 + 160y^6 + 240y^4 + 192y^2 + 64)
5. Simplify the resulting expanded expression, keeping like terms together:
(y^12)^2 + (12y^10)^2 + (60y^8)^2 + (-160y^6)^2 + (240y^4)^2 + (-192y^2)^2 + (64)^2
= y^24 + 144y^20 + 3600y^16 + 25600y^12 + 57600y^8 + 36864y^4 + 4096
Therefore, the final expanded and simplified expression is:
y^24 + 144y^20 + 3600y^16 + 25600y^12 + 57600y^8 + 36864y^4 + 4096