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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

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 power

True. 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.

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

1 answer

To determine whether the equation \( 3 - 5 \cdot (81\sqrt{4})^{3} - 4 = 3 - 6 \) is true or false, let’s simplify both sides step by step.

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

    • Calculate \( \sqrt{4} = 2 \).
    • Then, \( 81 \cdot 2 = 162 \).

  2. Now compute \( (162)^{3} \):

    • We do not need to compute \( 162^{3} \) directly for our purpose since we are going to substitute the results later in an expression.

  3. The left side of the equation:

    • The left side simplifies to: \[ 3 - 5 \cdot 162^{3} - 4 \]
    • Which can be further simplified to: \[ 3 - 4 - 5 \cdot 162^{3} = -1 - 5 \cdot 162^{3} \]

  4. Simplifying the right side:

    • The right side of the equation \( 3 - 6 \) simplifies to: \[ 3 - 6 = -3 \]

  5. Setting both sides equal:

    • We now compare \( -1 - 5 \cdot 162^{3} \) with \( -3 \) to verify the equation: \[ -1 - 5 \cdot 162^{3} \neq -3 \]
    • This is because \( 5 \cdot 162^{3} \) is a positive value, thus \( -1 - 5 \cdot 162^{3} \) is clearly less than \(-1\) and not equal to \(-3\).

Thus, the equation is False.

The correct justification in the context of the responses is:

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