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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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