How to solve the equation x^8 - 64 = 0?
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We have to solve x^8 - 64 = 0
Now 64 = 2^6
x^8 - 64 = 0
=> x^8 = 64
=> x^8 =...
(The entire section contains 39 words.)
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Let's recall first the formula for the difference of squares:
a^2 - b^2 =(a-b)(a+b)
Now,we'll put a^2 = x^8 = (x^4)^2 and b^2 = 64 = 8^2
We'll re-write the given equation, emphasizing on the difference of squares:
(x^4)^2 - 8^2 = (x^4 - 8)(x^4 + 8)
x^4 - 8 is also a difference o squares, whose terms are: a = x^2 and b = 2sqrt2
(x^4 - 8)(x^4 + 8) = (x^2 - 2sqrt2)(x^2 + 2sqrt2)(x^4 + 8)
Now, we'll solve the equation:
x^8 - 64 = 0
x^8 - 64 = (x^2 - 2sqrt2)(x^2 + 2sqrt2)(x^4 + 8)
(x^2 - 2sqrt2)(x^2 + 2sqrt2)(x^4 + 8) = 0
x^2 - 2sqrt2 = 0
x^2 = 2sqrt2
x1 = +sqrt(2sqrt2)
x2 = -sqrt(2sqrt2)
x^2 + 2sqrt2 = 0
x3 = +i*sqrt(2sqrt2)
x4 = -i*sqrt(2sqrt2)
The only real roots of the equation are {+sqrt(2sqrt2) ; -sqrt(2sqrt2)}.
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