What is the radical conjugate of $12-sqrt(5)$ ?

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Answer 1 · 2016-06-25T06:30:26

$12+sqrt(5)$

The conjugate of $12-sqrt(5)$ is $12+sqrt(5)$ .

This has the property that: $(12-sqrt(5))(12+sqrt(5))$ is rational:

$(12-sqrt(5))(12+sqrt(5)) = 12^2-(sqrt(5))^2 = 144-5 = 139$

Typical examples where you would use the conjugate would be:

When rationalising the denominator of a quotient.
When looking at zeros of a polynomial with rational (typically integer) coefficients.

For example:

$(2+3sqrt(5))/(12-sqrt(5))$

$=((2+3sqrt(5))(12+sqrt(5)))/((12-sqrt(5))(12+sqrt(5)))$

$=(24+2sqrt(5)+36sqrt(5)+15)/(144-5)$

$=(39+38sqrt(5))/139$

The simplest polynomial with rational coefficients and zero $12-sqrt(5)$ is:

$(x-(12-sqrt(5)))(x-(12+sqrt(5)))=x^2-24x+139$

What is the radical conjugate of 12-sqrt(5)12-sqrt(5) ?