Is the equation 6^1/2⋅(√6)^3=36 true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)

Question

Responses

False, because the exponents should be divided.

True, according to the Power of a Power Property.

True, according to the Product Property.

False, because the exponents should be multiplied.

Answer 1

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

First, we know that:

\[ 6^{1/2} = \sqrt{6} \]

Next, we can rewrite \( (\sqrt{6})^3 \):

\[ (\sqrt{6})^3 = (6^{1/2})^3 = 6^{(1/2) \cdot 3} = 6^{3/2} \]

Now, we can substitute these into the left-hand side of the equation:

\[ \sqrt{6} \cdot (\sqrt{6})^3 = 6^{1/2} \cdot 6^{3/2} \]

Using the Product Property of exponents, which states \( a^m \cdot a^n = a^{m+n} \), we can add the exponents:

\[ 6^{1/2} \cdot 6^{3/2} = 6^{(1/2 + 3/2)} = 6^{4/2} = 6^2 \]

And since:

\[ 6^2 = 36 \]

So the equation:

\[ 6^{1/2} \cdot (\sqrt{6})^3 = 36 \]

is true.

Thus, the correct response is:

True, according to the Product Property.