s the equation 3−5⋅(81√4)33−4=3−6

Question

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

True. (81−−√4)3=32
and 3−103−4=3−6
True. open paren 4th root of 81 close paren cubed is equal to 3 squared and the fraction with numerator 3 to the negative 10 power and denominator 3 to the negative 4 power is equal to 3 to the negative 6 power

True. The numerator simplifies to 3−2
and 3−23−4≠3−6
True. The numerator simplifies to 3 to the negative 2 power and the fraction with numerator 3 to the negative 2 power and denominator 3 to the negative 4 power is not equal to 3 to the negative 6 power

False. The numerator simplifies to 3−2
and 3−23−4≠3−6
False. The numerator simplifies to 3 to the negative 2 power and the fraction with numerator 3 to the negative 2 power and denominator 3 to the negative 4 power is not equal to 3 to the negative 6 power

False. (81−−√4)3
cannot be determined
False. open paren 4th root of 81 close paren cubed cannot be determined
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Answer 1

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

Starting with the left-hand side:

1. Evaluate \( (81\sqrt{4})^{3-4} \):

   • First calculate \( \sqrt{4} = 2 \).
   • Then \( 81\sqrt{4} = 81 \cdot 2 = 162 \).
   • Now, \( 3 - 4 = -1 \), so we find \( (162)^{-1} = \frac{1}{162} \).

2. Multiply by -5:

   • \( -5 \cdot \frac{1}{162} = -\frac{5}{162} \).

3. Combine with 3:

   • Thus, \( 3 - \frac{5}{162} = \frac{486}{162} - \frac{5}{162} = \frac{481}{162} \).

Now for the right-hand side:

1. Evaluate \( 3 - 6 = -3 \).

Now we compare:

• Left-hand side: \( \frac{481}{162} \)
• Right-hand side: \( -3 \) (which can be written as \( -\frac{486}{162} \))

Clearly, \( \frac{481}{162} \neq -\frac{486}{162} \).

Thus, the equation is false. The correct response is:

False. The numerator simplifies to \(3^{-2}\) and \(3^{-2} \neq 3^{-6}\).