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

2^1/2

√2

2^2

2

1 answer

Let's simplify the expression \((\sqrt{2})^3 \cdot 2^{-1/2}\) using properties of exponents.

  1. First, rewrite \(\sqrt{2}\) as \(2^{1/2}\): \[ (\sqrt{2})^3 = (2^{1/2})^3 \] Using the property of exponents \((a^m)^n = a^{mn}\), we get: \[ (2^{1/2})^3 = 2^{(1/2) \cdot 3} = 2^{3/2} \]

  2. Next, we include the second part of the expression: \[ 2^{3/2} \cdot 2^{-1/2} \] Using the property \(a^m \cdot a^n = a^{m+n}\), we combine the exponents: \[ 2^{3/2 - 1/2} = 2^{(3/2) - (1/2)} = 2^{(3/2 - 1/2)} = 2^{2/2} = 2^1 \] Therefore, we have: \[ 2^1 = 2 \]

Thus, the equivalent expression to \((\sqrt{2})^3 \cdot 2^{-1/2}\) is \(2\).

The correct answer is: \[ \boxed{2} \]