(sqrt[3](4)+2 sqrt(8))/(4 sqrt(8)-3 sqrt(6))

To simplify the expression \( \frac{\sqrt[3]{4}+2 \sqrt{8}}{4 \sqrt{8}-3 \sqrt{6}} \), we can start by simplifying the numerator and denominator separately.<br /><br />First, let's simplify the numerator:<br />\[ \sqrt[3]{4} + 2 \sqrt{8} \]<br /><br />We know that \( \sqrt[3]{4} = \sqrt[3]{2^2} = 2^{2/3} \) and \( \sqrt{8} = \sqrt{2^3} = 2^{3/2} \).<br /><br />So, the numerator becomes:<br />\[ 2^{2/3} + 2 \cdot 2^{3/2} \]<br /><br />Now, let's simplify the denominator:<br />\[ 4 \sqrt{8} - 3 \sqrt{6} \]<br /><br />We know that \( \sqrt{8} = \sqrt{2^3} = 2^{3/2} \).<br /><br />So, the denominator becomes:<br />\[ 4 \cdot 2^{3/2} - 3 \sqrt{6} \]<br /><br />Now, we can rewrite the original expression as:<br />\[{2^{2/3} + 2 \cdot 2^{3/2}}{4 \cdot 2^{3/2} - 3 \sqrt{6}} \]<br /><br />To simplify this expression further, we can factor out common terms in the numerator and denominator.<br /><br />In the numerator, we can factor out \( 2^{2/3} \):<br />\[ 2^{2/3} (1 + 2 \cdot 2^{1/2}) \]<br /><br />In the denominator, we can factor out \( 2^{3/2} \):<br />\[ 2^{3/2} (2 - 3 \sqrt{3/2}) \]<br /><br />Now, we can simplify the expression by canceling out common factors:<br />\[ \frac{2^{2/3} (1 + 2 \cdot 2^{1/2})}{2^{3/2} (2 - 3 \sqrt{3/2})} \]<br /><br />Finally, we can simplify the expression by canceling out the common factor of \( 2^{2/3} \) in the numerator and denominator:<br />\[ \frac{1 + 2 \cdot 2^{1/2}}{2 (2 - 3 \sqrt{3/2})} \]<br /><br />Therefore, the simplified form of the given expression is:<br />\[ \frac{1 + 2 \cdot 2^{1/2}}{2 (2 - 3 \sqrt{3/2})} \]