Simplify \frac{12}{\sqrt{3}} + \frac{4}{\sqrt{3} + 2}
Answer:
To simplify the expression \frac{12}{\sqrt{3}} + \frac{4}{\sqrt{3} + 2} we will first rationalize the denominators.
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Rationalize \frac{12}{\sqrt{3}} :
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Multiply the numerator and denominator by \sqrt{3} :
\frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12 \sqrt{3}}{3} = 4 \sqrt{3}
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Rationalize ( \frac{4}{\sqrt{3} + 2} ):
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Multiply the numerator and the denominator by the conjugate of the denominator ( \sqrt{3} - 2 ):
\frac{4}{\sqrt{3} + 2} \times \frac{\sqrt{3} - 2}{\sqrt{3} - 2} = \frac{4 (\sqrt{3} - 2)}{(\sqrt{3})^2 - (2)^2}-
Simplify the denominator:
(\sqrt{3})^2 - (2)^2 = 3 - 4 = -1 -
Therefore:
\frac{4 (\sqrt{3} - 2)}{-1} = -4 (\sqrt{3} - 2) = -4 \sqrt{3} + 8
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Combine the simplified terms:
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We now add the terms together:
4 \sqrt{3} + (-4 \sqrt{3} + 8) -
Simplifying:
4 \sqrt{3} - 4 \sqrt{3} + 8 = 8
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Final Answer:
The simplified form of the given expression is:
\boxed{8}