MHB Help with how to do this equation

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To solve the equation \dfrac{8 - \sqrt{27}}{2 \sqrt{3}}, rationalizing the denominator involves multiplying by \dfrac{\sqrt{3}}{\sqrt{3}}. After this step, the result should be in the form a + b\sqrt{3}. There is confusion about the nature of the numbers a and b, as the problem's wording affects the uniqueness of the solution. If a and b are real numbers, there are infinitely many solutions, but if they are specified as rational numbers, the solution is unique. Clarification on the problem's requirements is essential for accurate resolution.
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Hint: Rationalize the denominator: [math]\dfrac{8 - \sqrt{27}}{2 \sqrt{3}} = \dfrac{8 - \sqrt{27}}{2 \sqrt{3}} \cdot \dfrac{\sqrt{3}}{\sqrt{3}}[/math]

-Dan
 
I've done that but I'm not sure where to go from there.
 
ajk426 said:
I've done that but I'm not sure where to go from there.
You should have gotten a form that looks like [math]a + b \sqrt{3}[/math] so I don't know where the problem is. Please post what you got.

-Dan
 
When you multiplied both numerator and denominator by $\sqrt{3}$, what did you get?
 
Did the problem really say that a and b are real numbers? If so there are infinitely many correct answers. IF the problem had specified "rational number" then there would be a unique answer.
 

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