Using the exponent properties, which of the following expressions is equivalent to (radical notation 2)^3 ⋅ 2^−1/2

To simplify the expression \((\sqrt{2})^3 \cdot 2^{-\frac{1}{2}}\), we can first convert the radical notation into exponent form:

1. The expression \(\sqrt{2}\) can be written as \(2^{\frac{1}{2}}\).
2. Therefore, \((\sqrt{2})^3\) becomes:
\[
(2^{\frac{1}{2}})^3 = 2^{\frac{1}{2} \cdot 3} = 2^{\frac{3}{2}}.
\]

Next, we rewrite the entire expression:

\[
2^{\frac{3}{2}} \cdot 2^{-\frac{1}{2}}.
\]

Now, according to the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:

\[
2^{\frac{3}{2} - \frac{1}{2}} = 2^{\frac{3 - 1}{2}} = 2^{\frac{2}{2}} = 2^1.
\]

Therefore, the simplified expression is:

\[
2^1 = 2.
\]

So, the equivalent expression is \(2\).