To simplify the expression \((\sqrt{2})^3 \cdot 2^{-\frac{1}{2}}\), we will use the properties of exponents.
First, recognize that \(\sqrt{2}\) can be rewritten as \(2^{\frac{1}{2}}\): \[ (\sqrt{2})^3 = (2^{\frac{1}{2}})^3 \] Applying the power of a power rule \((a^m)^n = a^{m \cdot n}\), we get: \[ (2^{\frac{1}{2}})^3 = 2^{\frac{3}{2}} \]
Now, let's rewrite the entire expression: \[ 2^{\frac{3}{2}} \cdot 2^{-\frac{1}{2}} \] Next, apply the property of exponents that states \(a^m \cdot a^n = a^{m+n}\): \[ 2^{\frac{3}{2} - \frac{1}{2}} = 2^{\frac{3-1}{2}} = 2^{\frac{2}{2}} = 2^1 \]
Thus, we find that: \[ 2^1 = 2 \]
The equivalent expression to \((\sqrt{2})^3 \cdot 2^{-\frac{1}{2}}\) is: \[ \boxed{2} \]