write in the form a + b root 3---Mathematical Expression:
The expression is a complex fraction.
Numerator: $\frac{\sqrt{3}}{2} + 1$
Denominator: $1 - \frac{\sqrt{3}}{2}$
Full expression:
$$ \frac{\frac{\sqrt{3}}{2} + 1}{1 - \frac{\sqrt{3}}{2}} $$
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We have a complex fraction with square root of 3 terms. Our goal is to simplify this expression and write it in the standard form a plus b times square root of 3, where a and b are rational numbers.
The first step is to rewrite both the numerator and denominator with common denominators. For the numerator, we convert 1 to two halves, giving us square root 3 plus 2, all over 2. For the denominator, we convert 1 to two halves, giving us 2 minus square root 3, all over 2.
Now we simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator. This gives us square root 3 plus 2, all over 2 minus square root 3.
To rationalize the denominator, we multiply both numerator and denominator by the conjugate 2 plus square root 3. Expanding the numerator gives us 7 plus 4 square root 3. The denominator becomes 4 minus 3, which equals 1. Therefore, our final answer is 7 plus 4 square root 3.
We have successfully simplified the complex fraction. The 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 completes our solution.