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.(1 point)

Question

Responses

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

True, but the equation should be simplified using the Negative Exponent Rule.
True, but the equation should be simplified using the Negative Exponent Rule.

False. There is an error in applying the Product Property.
False. There is an error in applying the Product Property.

True. All rules have been applied correctly.

Answer 1

To evaluate the equation \( 3 - 5 \cdot (81 \sqrt{4})^{33 - 4} = 3 - 6 \), let's simplify both sides step by step.

1. Simplify the left side:

   • Calculate \( 33 - 4 \): \[ 33 - 4 = 29 \]

   • We need to compute \( (81 \sqrt{4})^{29} \):

     • Calculate \( \sqrt{4} \) which equals \( 2 \), so: \[ 81 \sqrt{4} = 81 \cdot 2 = 162 \]

     • Now we have: \[ (162)^{29} \]

   • Therefore, the left side becomes: \[ 3 - 5 \cdot (162)^{29} \]

2. Simplify the right side:

   • Calculate \( 3 - 6 \): \[ 3 - 6 = -3 \]

Now we can compare both sides:

• Left side: \( 3 - 5 \cdot (162)^{29} \)
• Right side: \( -3 \)

Clearly, \( 3 - 5 \cdot (162)^{29} \) cannot equal \( -3 \) since \( 5 \cdot (162)^{29} \) is a very large positive number. Therefore, the left side is not equal to the right side.

Thus, the equation is false.

The justification is: False. There is an error in calculation using the Quotient Property. (Note: The response should have indicated there was a mistake in the simplification or understanding of the original expression rather than specifically stating "Quotient Property" since we didn't use that property here; however, this option is the closest in indicating a calculation error.)