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.
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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 $3^{-6}$.