Radicals & Rational Exponents Quick Check

Question

5 of 55 of 5 Items
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. (81−−√4)3
cannot be determined.
False. open paren 4th root of 81 close paren cubedcannot be determined.

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.

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

Answer 1

To determine whether the equation \( 3^{-5} \cdot (81^{\sqrt{4}})^{3} \cdot 3^{-4} = 3^{-6} \) is true or false, we will simplify each side.

1. Simplifying the left side:

   • First, calculate \( \sqrt{4} \), which is \( 2 \), thus \( 81^{\sqrt{4}} = 81^{2} \).
   • Since \( 81 = 3^{4} \), we can write: \[ 81^{2} = (3^{4})^{2} = 3^{8} \]
   • Now plugging this back into the left side: \[ 3^{-5} \cdot (3^{8})^{3} \cdot 3^{-4} = 3^{-5} \cdot 3^{24} \cdot 3^{-4} \]
   • Using the laws of exponents, we combine: \[ 3^{-5 + 24 - 4} = 3^{15} \]

2. Simplifying the right side:

   • The right side is \( 3^{-6} \).

3. Comparing both sides:

   • We have \( 3^{15} \) on the left and \( 3^{-6} \) on the right. Clearly, these two expressions are not equal.

Thus, the statement \( 3^{-5} \cdot (81^{\sqrt{4}})^{3} \cdot 3^{-4} = 3^{-6} \) is false.

Conclusion: The correct response is: False. The numerator simplifies to \( 3^{-2} \) and the fraction with numerator \( 3^{-2} \) and denominator \( 3^{-4} \) is not equal to \( 3^{-6} \).