Question
Radicals & Rational Exponents Quick Check
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
Answers
GPT-4o mini
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 \( 3^{-6} \).**
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 \( 3^{-6} \).**