write in the form a + b root 3 but using conjugating in the first step so it becomes the equation in the image multiplied by the conjucate over the conjugate of the denominator---$\frac{\frac{\sqrt{3}}{2} + 1}{1 - \frac{\sqrt{3}}{2}}$
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To rationalize this complex fraction, we first identify the denominator which contains a square root term. The denominator is 1 minus square root 3 over 2. To eliminate the square root from the denominator, we multiply both numerator and denominator by the conjugate of the denominator, which is 1 plus square root 3 over 2.
Now we expand the numerator. We notice that the numerator becomes the square of 1 plus square root 3 over 2. Using the perfect square formula, this equals 1 squared plus 2 times 1 times square root 3 over 2, plus square root 3 over 2 squared. This simplifies to 1 plus square root 3 plus 3 over 4, which equals 7 over 4 plus square root 3.
For the denominator, we use the difference of squares formula. When we multiply 1 minus square root 3 over 2 by 1 plus square root 3 over 2, we get 1 squared minus square root 3 over 2 squared. This equals 1 minus 3 over 4, which simplifies to 1 over 4.
Now we have the fraction 7 over 4 plus square root 3, all divided by 1 over 4. To divide by a fraction, we multiply by its reciprocal. So we multiply by 4. Distributing the 4, we get 7 over 4 times 4 plus square root 3 times 4, which equals 7 plus 4 square root 3. This is our final answer in the form a plus b square root 3.
In summary, we have successfully rationalized the complex fraction by multiplying by the conjugate of the denominator. Our original expression equals 7 plus 4 square root 3, which is in the required form a plus b square root 3, where a equals 7 and b equals 4. This method of rationalization is essential for simplifying expressions containing square roots in denominators.