Question

Using exponent properties which of the following expressions is equivalent to ( square root 2)^3•2^-1/2
2^1/2
Square Root 2
2^2

Answer 1

To simplify the expression \((\sqrt{2})^3 \cdot 2^{-1/2}\), we can use the properties of exponents.

First, recall that \(\sqrt{2}\) can be expressed as \(2^{1/2}\). Therefore, we can rewrite \((\sqrt{2})^3\) as:

\[
(\sqrt{2})^3 = (2^{1/2})^3 = 2^{(1/2) \cdot 3} = 2^{3/2}
\]

Now, substituting this back into the original expression gives us:

\[
2^{3/2} \cdot 2^{-1/2}
\]

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

\[
2^{3/2 - 1/2} = 2^{(3/2 - 1/2)} = 2^{2/2} = 2^1
\]

Now, \(2^1\) is simply 2.

So the simplified expression is equivalent to:

1. \(2^{1/2} = \sqrt{2}\)
2. \(2^1\)
3. \(\sqrt{2}\)
4. \(2^2\)

### Final Answer:
**2**