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

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

Answer 1

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

Starting with the left side:

1. Calculate \( \sqrt{4} \): \[ \sqrt{4} = 2 \]

2. Substitute back: \[ 81\sqrt{4} = 81 \cdot 2 = 162 \]

3. Now we have: \[ (81\sqrt{4})^{3 - 4} = (162)^{-1} = \frac{1}{162} \]

4. Substitute this into the left side of the equation: \[ 3 - 5 \cdot \frac{1}{162} \]

5. Calculate \( 5 \cdot \frac{1}{162} \): \[ 5 \cdot \frac{1}{162} = \frac{5}{162} \]

6. Thus, the left side becomes: \[ 3 - \frac{5}{162} \]

To simplify \( 3 - \frac{5}{162} \), write \( 3 \) as a fraction: \[ 3 = \frac{486}{162} \] Now subtract: \[ \frac{486}{162} - \frac{5}{162} = \frac{486 - 5}{162} = \frac{481}{162} \]

Now, let's evaluate the right side: \[ 3 - 6 = -3 \]

So we compare: Left side: \( \frac{481}{162} \) Right side: \( -3 \)

Since \( \frac{481}{162} \neq -3 \), the equation is false.

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