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)ResponsesTrue. The numerator simplifies to 3−2and 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.True. (81−−√4)3=32 and 3−103−4=3−6True. 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 powerFalse. (81−−√4)3cannot be determined.

Question

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)ResponsesTrue. The numerator simplifies to 3−2and 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.True. (81−−√4)3=32 and 3−103−4=3−6True. 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 powerFalse. (81−−√4)3cannot be determined.

GPT-4o mini · Answer

To determine whether the equation $ 3^{-5 \cdot (\sqrt[4]{81})^3} \cdot 3^{-4} = 3^{-6} $ is true or false, let’s simplify the left-hand side. Firstly, calculate $ \sqrt[4]{81} $: $$ \sqrt[4]{81} = \sqrt[4]{3^4} = 3 $$ Now we substitute this back into our equation: $$ 3^{-5 \cdot (3)^3} \cdot 3^{-4} $$ Next, calculate $ (3)^3 $: $$ (3)^3 = 27 $$ Thus, our expression simplifies to: $$ 3^{-5 \cdot 27} \cdot 3^{-4} = 3^{-135} \cdot 3^{-4} $$ Using the properties of exponents to combine the exponents: $$ 3^{-135 - 4} = 3^{-139} $$ Now we need to compare $ 3^{-139} $ with $ 3^{-6} $. Clearly, $$ 3^{-139} \neq 3^{-6} $$ Thus, the equation $ 3^{-5 \cdot (\sqrt[4]{81})^3} \cdot 3^{-4} = 3^{-6} $ is **false**. The correct justification is: **False. The numerator simplifies to $3^{-139}$ and $3^{-139} \neq 3^{-6}$.**