The number $\sqrt{63-36 \sqrt{3}}$ can be expressed as $x+y \sqrt{3}$ for some integers $x$ and $y$. What is the value of $x y$ ?

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The number 63 - 363 can be expressed as x + y 3 for some integers x and y . What is the value of x y ?

Explanation	Calculations	Help
$We let integers x and y satisfy the given form.$	$63 - 363 = x + y 3$
$We square both sides and expand the result.$	$63 - 363 = x^{2} + 3 y^{2} + 2 x y 3$	Theory & Tactics Method Card ALG4-A Click to view full details
$We separate the equation into its rational part and its irrational part.$	$x^{2} + 3 y^{2} = 63 2 x y = - 36$
$We solve the second equation for the product of x and y.$	$x y = - 18$

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The number 63 - 363 can be expressed as x + y 3 for some integers x and y . What is the value of x y ?

Explanation	Calculations	Help
$We let integers x and y satisfy the given form.$	$63 - 363 = x + y 3$
$We square both sides and expand the result.$	$63 - 363 = x^{2} + 3 y^{2} + 2 x y 3$	Theory & Tactics Method Card ALG4-A Click to view full details
$We separate the equation into its rational part and its irrational part.$	$x^{2} + 3 y^{2} = 63 2 x y = - 36$
$We solve the second equation for the product of x and y.$	$x y = - 18$

Scroll horizontally to view all columns

-18