Is the equation 3−5⋅(81√4)33−4=3−6 true or false? Select the response that correctly answers the question and provides the appropriate justification. Responses True. All rules have been applied correctly. True. All rules have been applied correctly. - no response given False. There is an error in calculation using the Quotient Property. False. There is an error in calculation using the Quotient Property. - no response given False. There is an error in applying the Product Property. False. There is an error in applying the Product Property. - no response given True, but the equation should be simplified using the Negative Exponent Rule. True, but the equation should be simplified using the Negative Exponent Rule. - no response given

Answer 1

To determine whether the equation \( 3 - 5 \cdot (81\sqrt{4})^{3-4} = 3 - 6 \) is true or false, let’s simplify both sides step by step.

1. Simplify the left side \( 3 - 5 \cdot (81\sqrt{4})^{3-4} \):

   • Calculate \( \sqrt{4} = 2 \), so \( 81\sqrt{4} = 81 \cdot 2 = 162 \).
   • Now, evaluate the exponent: \( 3 - 4 = -1 \).
   • So, we have \( (162)^{-1} = \frac{1}{162} \).
   • Now plug this back into the equation: \[ 3 - 5 \cdot \frac{1}{162} = 3 - \frac{5}{162}. \]
   • To combine the terms, find a common denominator: \[ 3 = \frac{486}{162} \Rightarrow 3 - \frac{5}{162} = \frac{486 - 5}{162} = \frac{481}{162}. \]

2. Simplify the right side \( 3 - 6 \):

   • This is straightforward: \( 3 - 6 = -3 \).

Now we compare the left side to the right side:

• Left side: \( \frac{481}{162} \)
• Right side: \( -3 = -\frac{486}{162} \)

Since \( \frac{481}{162} \neq -\frac{486}{162} \), the left side does not equal the right side.

Therefore, we conclude that the statement is False: There is an error in calculation using the Quotient Property.

The conclusion is: False. There is an error in calculation using the Quotient Property.