How does (4 + 1/(x^2) + 4x^2)^1/2 = (2x^2 + 1)/x?

Question

need some help simplifying this, thanks

Answer 1

(4 + 1/(x^2) + 4x^2)^1/2
= [(4x^2 + 1 + 4x^4)/x^2]^1/2
= [(4x^4 + 4x^2 + 1)/x^2]^1/2
= [(2x^2 + 1) / x^2]^1/2
= (2x^2 + 1) / x


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Answer 2

You can multiply and divide the 4 and the 4x^2 each by x^2, factor out the sqrt(1/x^2), and hope the remaining numerator can be converted to the numerator shown.

Answer 3

[(2x^2 + 1)/x]^2 = [(2x^2)^2 + 2 * 2x^2 + 1]/x^2 =
= (4x^4 + 4x^2 + 1)/x^2 = 4x^2 + 4 + 1/(x^2)