Discussion
If \(\frac{5 - \sqrt{12}}{7 + \sqrt{48}} = a + b\sqrt{3}\), then what is the value of \(\sqrt{a + b}\) ?
Answer: B
The value of the square root is 5. First write the denominator in simple form: the square root of 48 is 4 times the square root of 3 and the square root of 12 is 2 times the square root of 3. Then multiply the top and bottom by the conjugate of the denominator.
\[ \frac{5 - 2\sqrt{3}}{7 + 4\sqrt{3}} \]
\[ \times \frac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} \]
\[ = \frac{35 - 20\sqrt{3} - 14\sqrt{3} + 24}{49 - 48} \]
\[ = 59 - 34\sqrt{3} \]
So a = 59 and b = -34, and a + b = 25. Then the square root of (a + b) is the square root of 25, which is 5. The options 4, 3 and 2 do not come from this value of a + b.
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