Question

the equation of lowest degree with rational coefficients having a root (root2+root3) then the equation is_​

Answer 1

SOLUTION

TO DETERMINE

The equation of lowest degree with rational coefficients having a root

$\sf{ \sqrt{( \sqrt{2} + \sqrt{3} )} }$

EVALUATION

Let

$\sf{ x = \sqrt{( \sqrt{2} + \sqrt{3} )} }$

$\sf{ \implies \: {x}^{2} =( \sqrt{2} + \sqrt{3} )}$

$\sf{ \implies \: {x}^{4} =4 + 3 + 2 \times \sqrt{2} \times \sqrt{3} }$

$\sf{ \implies \: {x}^{4} =7+ 2\sqrt{6} }$

$\sf{ \implies \: {x}^{4} - 7 = 2\sqrt{6} }$

$\sf{ \implies \: {({x}^{4} - 7)}^{2} = {(2\sqrt{6} )}^{2} }$

$\sf{ \implies \: {({x}^{4} - 7)}^{2} = 24}$

$\sf{ \implies \: {x}^{8} - 14 {x}^{4} + 49= 24}$

$\sf{ \implies \: {x}^{8} - 14 {x}^{4} + 25= 0}$

Which is the required equation with rational coefficients

Now degree of the equation = 8

FINAL ANSWER

The equation of lowest degree with rational coefficients having given root

$\sf{ {x}^{8} - 14 {x}^{4} + 25= 0}$

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