How to factor the difference a^4-b^16?
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We have to factor a^4 - b^16
we use x^2 - y^2 = (x +y)(x - y)
a^4 - b^16
=> (a^2 - b^8)(a^2 + b^8)
=> (a - b^4)(a + b^4)(a^2 + b^8)
Therefore a^4 - b^16 = (a - b^4)(a + b^4)(a^2 + b^8)
We'll use the power property:
a^4 = a^2*2 = (a^2)^2
b^16 = b^2*8 = (b^8)^2
we'll write the terms in this way:
(a^2)^2 - (b^8)^2
We notice that we've get a difference of squares:
x^2 - y^2 = (x-y)(x+y)
We'll put x = a^2 and y = b^8:
(a^2)^2 - (b^8)^2 = (a^2 - b^8)(a^2 + b^8) (1)
We notice that the factor a^2 - b^8 is also a difference of squares:
a^2 - b^8 = (a)^2 - (b^4)^2
(a)^2 - (b^4)^2 = (a - b^4)(a + b^4) (2)
We'll substitute (2) in (1):
a^4-b^16 = (a - b^4)(a + b^4)(a^2 + b^8)
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