Is the equation 3−5⋅(81√4)33−4=3−6

Question

3
−
5
⋅
(
81
4
)
3
3
−
4
=
3
−
6
true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

True. The numerator simplifies to 3−2
3
−
2
and 3−23−4≠3−6
3
−
2
3
−
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
3
−
2
and 3−23−4≠3−6
3
−
2
3
−
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
(
81
4
)
3
cannot be determined.
False. open paren 4th root of 81 close paren cubedcannot be determined.

True. (81−−√4)3=32
(
81
4
)

Answer 1

To determine whether the equation

\[ 3 - 5 \cdot (81 \sqrt{4})^{\frac{3}{3}} - 4 = 3 - 6 \]

is true or false, we first need to simplify the left side of the equation.

Let's simplify step by step:

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

2. Replace \(\sqrt{4}\) in the expression \(81 \sqrt{4}\): \[ 81 \cdot 2 = 162 \]

3. Now simplify \((162)^{\frac{3}{3}}\): \[ (162)^{\frac{3}{3}} = 162 \]

4. Now we can substitute back into the original expression: \[ 3 - 5 \cdot 162 - 4 \]

5. Calculate \(5 \cdot 162\): \[ 5 \cdot 162 = 810 \]

6. Now substitute this value into the expression: \[ 3 - 810 - 4 = 3 - 810 - 4 = 3 - 814 = -811 \]

Now let's find the right side of the equation:

1. Calculate \(3 - 6\): \[ 3 - 6 = -3 \]

Now we can rewrite the entire equation as:

\[ -811 \neq -3 \]

Thus, the equation is not valid. The answer is:

False. The numerator simplifies to \(3^{-2}\) and \(3^{-2} / 3^{-4} \neq 3^{-6}\).