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

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.

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

True. All rules have been applied correctly.
True. All rules have been applied correctly.

Answer 1

To determine whether the equation \( 3 - 5 \cdot (81 \sqrt{4})^{33 - 4} = 3 - 6 \) is true or false, we need to simplify both sides of the equation.

First, simplify the expression \( 81 \sqrt{4} \):

• We know \( \sqrt{4} = 2 \).
• Thus, \( 81 \sqrt{4} = 81 \cdot 2 = 162 \).

Now substitute this back into the equation: \[ 3 - 5 \cdot (162)^{33 - 4} = 3 - 6 \]

Next, calculate \( 33 - 4 \): \[ 33 - 4 = 29 \] Thus, we can rewrite \( (162)^{33 - 4} \) as \( (162)^{29} \): \[ 3 - 5 \cdot (162)^{29} = 3 - 6 \]

Now, we need to simplify the right side: \[ 3 - 6 = -3 \]

Now the equation reads: \[ 3 - 5 \cdot (162)^{29} = -3 \]

To solve for \( 5 \cdot (162)^{29} \): \[ 3 - (-3) = 5 \cdot (162)^{29} \] \[ 3 + 3 = 5 \cdot (162)^{29} \] \[ 6 = 5 \cdot (162)^{29} \]

Clearly, \( 162^{29} \) is a much larger number than \( 6/5 \), meaning that: \[ 5 \cdot (162)^{29} \neq 6 \]

Thus, the original equation is false.

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

(Note: The specific mention of "Quotient Property" might not apply here, but since the equation is false due to the calculations, this choice aligns more closely with the fact that there was an error in evaluating the overall expression.)