Question

Is the equation 6 ^ (1/2) * (sqrt(6)) ^ 3 = 36 true or false? Select the response that correctly answers the question and provides the appropriate justification. (1 point) False, because the exponents should be multiplied. True, according to the Power of a Power Property. True, according to the Product Property. False, because the exponents should be divided.

Answer 1

To evaluate the equation \( 6^{(1/2)} \cdot (\sqrt{6})^3 = 36 \), let's simplify the left side step by step.

1. First, recall that \(\sqrt{6} = 6^{1/2}\). Therefore, \((\sqrt{6})^3 = (6^{1/2})^3\).
2. According to the Power of a Power Property \((a^m)^n = a^{m \cdot n}\), we can rewrite \((\sqrt{6})^3\) as: \[ (6^{1/2})^3 = 6^{(1/2) \cdot 3} = 6^{3/2} \]
3. Now, we have: \[ 6^{(1/2)} \cdot 6^{(3/2)} \]
4. According to the Product Property \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents: \[ 6^{(1/2) + (3/2)} = 6^{(1/2 + 3/2)} = 6^{4/2} = 6^2 \]
5. Simplifying \(6^2\) gives us: \[ 6^2 = 36 \]

Thus, the original statement is true: \(6^{(1/2)} \cdot (\sqrt{6})^3 = 36\).

The correct response is: True, according to the Power of a Power Property.