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Simplify: \( \sqrt{432}-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3} \)
Given:
\( \sqrt{432}-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3} \)
To do:
We have to simplify \( \sqrt{432}-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3} \).
Solution:
$\sqrt{432}-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3}=\sqrt{36\times4\times3}-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3}$
$=6\times2\sqrt3-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3}$
$=12\sqrt3-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3}$
$=\frac{2\sqrt3\times12\sqrt3-5+2\sqrt3\times4\sqrt3}{2\sqrt3}$
$=\frac{24\times3-5+8\times3}{2\sqrt3}$
$=\frac{72-5+24}{2\sqrt3}$
$=\frac{91}{2\sqrt3}$
$=\frac{91\sqrt3}{2\sqrt3\times\sqrt3}$
$=\frac{91\sqrt3}{6}$
Therefore,
$\sqrt{432}-\frac{5}{2} \sqrt{\frac{1}{3}}+4 \sqrt{3}=\frac{91\sqrt3}{6}$.