using the exponent properties, which of the following is equivalent to (2 under a square root) with an exponent of 3 x 2 with an exponent of -1/2? 2 with an exponent of 3/2, 2, 2 with an exponent of 2, or 2 under a square root?

Answer 1

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

1. Rewrite \(\sqrt{2}\) as \(2^{1/2}\): \[ (\sqrt{2})^{3} = (2^{1/2})^{3} \]

2. Apply the power of a power property \( (a^m)^n = a^{m \cdot n} \): \[ (2^{1/2})^{3} = 2^{(1/2) \cdot 3} = 2^{3/2} \]

3. Now we combine this with \(2^{-1/2}\): \[ 2^{3/2} \cdot 2^{-1/2} \]

4. Using the property \(a^m \cdot a^n = a^{m+n}\): \[ 2^{3/2 - 1/2} = 2^{(3/2) - (1/2)} = 2^{2/2} = 2^{1} \]

Therefore, the expression simplifies to \(2\).

So the equivalent expression is: \(\mathbf{2}\).