(8-i)^2(8+i)^2
(8-i)(8+i) (8-i)(8+i)
[ 64 + 1 ] [ 64 + 1 ]
65 * 65 = 65^2 = 4225
(i^2-64)^2
Kimberly,
i^2 = -1
(-65)^2 = 4225
To simplify the expression (8-i)^2(8+i)^2, we can use the properties of complex numbers.
First, let's expand each square individually.
(8-i)^2 = (8-i)(8-i)
= 8(8) + 8(-i) - i(8) - i(-i)
= 64 - 8i - 8i + i^2
= 64 - 16i + i^2
Similarly,
(8+i)^2 = (8+i)(8+i)
= 8(8) + 8(i) + i(8) + i(i)
= 64 + 8i + 8i + i^2
= 64 + 16i + i^2
Now, multiply the expanded expressions together:
(64 - 16i + i^2)(64 + 16i + i^2)
To multiply these binomials, we can use the FOIL method (First, Outer, Inner, Last):
FOIL method:
First: Multiply the first terms of each binomial.
Outer: Multiply the outer terms of each binomial.
Inner: Multiply the inner terms of each binomial.
Last: Multiply the last terms of each binomial.
(64 * 64) + (64 * 16i) + (64 * i^2) + (-16i * 64) + (-16i * 16i) + (-16i * i^2) + (i^2 * 64) + (i^2 * 16i) + (i^2 * i^2)
Simplifying further:
4096 + 1024i - 64i - 256 + 256i^2 - 16i^2 + 64i^2 + 16i^3 + i^4
Combine like terms:
4096 - 64 - 256 + 256i^2 + 16i^2 + 64i^2 + 16i^3 + i^4
Now, let's simplify the powers of "i":
i^2 = -1 (since i^2 is defined as -1)
i^3 = i^2 * i = (-1) * i = -i
i^4 = (i^2)^2 = (-1)^2 = 1
Substituting these values into our expression:
4096 - 64 - 256 + 256(-1) + 16(-1) + 64(-1) + 16(-i) + 1
Simplifying further:
4096 - 64 - 256 - 256 - 16 - 64 - 16i + 1
Combine like terms:
3701 - 80i
Therefore, (8 - i)^2(8 + i)^2 simplifies to 3701 - 80i.