To simplify the expression \( \frac{8\sqrt[3]{2}}{\sqrt{}} \), we need to clarify the expression since it appears incomplete. The square root in the denominator should contain a term, let's assume it's \(\sqrt{2}\), and we have:
\[ \frac{8\sqrt[3]{2}}{\sqrt{2}} \]
Now, we can express the square root using exponents:
\[ \sqrt{2} = 2^{1/2} \] \[ \sqrt[3]{2} = 2^{1/3} \]
Now we can rewrite the expression:
\[ \frac{8 \cdot 2^{1/3}}{2^{1/2}} \]
Next, we simplify the \(8\) using exponent notation:
\[ 8 = 2^3 \]
So now we have:
\[ \frac{2^3 \cdot 2^{1/3}}{2^{1/2}} \]
Using the properties of exponents, \(a^m \cdot a^n = a^{m+n}\), we can combine \(2^3\) and \(2^{1/3}\):
\[ 2^{3 + \frac{1}{3}} = 2^{\frac{9}{3} + \frac{1}{3}} = 2^{\frac{10}{3}} \]
Now, we substitute this back into our expression and apply the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[ \frac{2^{\frac{10}{3}}}{2^{\frac{1}{2}}} = 2^{\frac{10}{3} - \frac{1}{2}} \]
To perform the subtraction, we need a common denominator, which is \(6\):
\[ \frac{10}{3} = \frac{20}{6} \] \[ \frac{1}{2} = \frac{3}{6} \]
Now calculate:
\[ \frac{20}{6} - \frac{3}{6} = \frac{17}{6} \]
Thus, we have:
\[ 2^{\frac{17}{6}} \]
So, the equivalent expression for \( \frac{8\sqrt[3]{2}}{\sqrt{2}} \) is:
\[ 2^{\frac{17}{6}} \]