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. (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
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 cubedcannot be determined.
True. The numerator simplifies to 3−2
and 3−23−4≠3−6.

Answer 1

To evaluate the equation \(3 - 5 \cdot (81^{\sqrt{4}})^{3} - 4 = 3 - 6\), let's break it down step by step.

1. First, simplify \(81^{\sqrt{4}}\). Since \(\sqrt{4} = 2\), we have: \[ 81^{\sqrt{4}} = 81^2 \] And \(81 = 3^4\), so: \[ 81^2 = (3^4)^2 = 3^{8} = 6561 \]

2. Next, calculate \((81^{\sqrt{4}})^{3}\): \[ (81^2)^{3} = 81^{6} = (3^4)^6 = 3^{24} \]

3. Now, we substitute back into the original equation: \[ 3 - 5 \cdot 3^{24} - 4 = 3 - 6 \]

4. Simplifying both sides, we find: Left side: \(3 - 4 - 5 \cdot 3^{24} = -1 - 5 \cdot 3^{24}\)
   Right side: \(3 - 6 = -3\)

Now consider the equality: \(-1 - 5 \cdot 3^{24} \neq -3\) since \(5 \cdot 3^{24}\) is a large positive number, making the left side a greatly negative number.

Thus, putting this all together, the statement \(3 - 5 \cdot (81^{\sqrt{4}})^{3} - 4 = 3 - 6\) is False.

Among the options given, the correct justification can be selected as: 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.