Is the equation 612⋅(6–√)3=36

Question

true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

True, according to the Power of a Power Property.
True, according to the Power of a Power Property.

True, according to the Product Property.
True, according to the Product Property.

False, because the exponents should be multiplied.
False, because the exponents should be multiplied.

False, because the exponents should be divided.
False, because the exponents should be divided.
Remaining Attempts : 3

Answer 1

To evaluate the equation \( 612 \cdot (6 - \sqrt{3})^3 = 36 \), we can first simplify the left-hand side.

1. Calculate \((6 - \sqrt{3})^3\):

   Using the binomial expansion: \[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \] where \( a = 6 \) and \( b = \sqrt{3} \):

   • \( a^3 = 6^3 = 216 \)
   • \( 3a^2b = 3 \cdot 6^2 \cdot \sqrt{3} = 3 \cdot 36 \cdot \sqrt{3} = 108\sqrt{3} \)
   • \( 3ab^2 = 3 \cdot 6 \cdot (\sqrt{3})^2 = 3 \cdot 6 \cdot 3 = 54 \)
   • \( b^3 = (\sqrt{3})^3 = 3\sqrt{3} \)

   Combining these: \[ (6 - \sqrt{3})^3 = 216 - 108\sqrt{3} + 54 - 3\sqrt{3} = 270 - 111\sqrt{3} \]

2. Plug this back into the equation: \[ 612 \cdot (270 - 111\sqrt{3}) = 36 \]

3. Consider the left side: \[ 612 \cdot (270 - 111\sqrt{3}) \text{ is not likely to equal 36.} \]

Without calculating the exact values (as they would be complicated), it is clear that multiplying a large number (612) by a cubic expression will not yield 36.

Therefore, the statement is False, because the left side would yield a value far greater than 36.

The correct response is: False, because the exponents should be multiplied.