Ask a New Question
Menu

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)
Responses

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=3−6
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. (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
.
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.
Skip to navigation

1 answer

To assess whether the equation \( 3^{-5 \cdot (81^{1/4})^3} - 4 = 3^{-6} \) is true or false, let's break it down step by step.

  1. Calculate \( 81^{1/4} \): \[ 81 = 3^4 \implies 81^{1/4} = (3^4)^{1/4} = 3^{4 \cdot \frac{1}{4}} = 3^1 = 3 \]

  2. Now calculate \( (81^{1/4})^3 \): \[ (81^{1/4})^3 = 3^3 = 27 \]

  3. Calculate \( -5 \cdot (81^{1/4})^3 \): \[ -5 \cdot 27 = -135 \]

  4. Evaluate the expression \( 3^{-135} - 4 \): So the original left-hand side becomes \( 3^{-135} - 4 \).

  5. Right-hand side is \( 3^{-6} \).

  6. Compare the two sides: The left-hand side \( 3^{-135} - 4 \) clearly does not equal \( 3^{-6} \) because \( 3^{-135} \) is an extremely small number compared to \( 4 \), and certainly not equal to \( 3^{-6} \).

Given the calculations and comparisons, the statement is:

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

However, the valid justification option you provided is incorrectly stating the simplification process. In this case, it's best to state that the entire left-hand side is not equal to the right-hand side based on our earlier steps, concluding the overall equation to be false.