Fleming Donnelly
11/17/2023 · Junior High School
n. 193 PAG. \( 3^{6} \) \( \left\{\left[\left(2^{3}\right)^{5}: 2^{11}-3^{2}-2^{2}\right]^{5} \cdot 3^{3}\right\}:\left(3^{6}\right)^{12}= \)
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Tutor-Verified Answer
Step-by-step Solution
To solve the expression
\[
\left\{\left[\left(2^{3}\right)^{5}: 2^{11}-3^{2}-2^{2}\right]^{5} \cdot 3^{3}\right\}:\left(3^{6}\right)^{12}
\]
we will break it down step by step.
1. **Calculate \( (2^3)^5 \)**:
\[
(2^3)^5 = 2^{3 \cdot 5} = 2^{15}
\]
2. **Calculate \( 2^{11} - 3^2 - 2^2 \)**:
- First, calculate \( 3^2 \) and \( 2^2 \):
\[
3^2 = 9, \quad 2^2 = 4
\]
- Now substitute these values:
\[
2^{11} - 9 - 4 = 2^{11} - 13
\]
3. **Now we have the expression**:
\[
\left[2^{15} : (2^{11} - 13)\right]^5 \cdot 3^3
\]
4. **Calculate the division**:
\[
2^{15} : (2^{11} - 13) = \frac{2^{15}}{2^{11} - 13}
\]
5. **Raise to the power of 5**:
\[
\left(\frac{2^{15}}{2^{11} - 13}\right)^5 = \frac{(2^{15})^5}{(2^{11} - 13)^5} = \frac{2^{75}}{(2^{11} - 13)^5}
\]
6. **Now multiply by \( 3^3 \)**:
\[
\frac{2^{75} \cdot 3^3}{(2^{11} - 13)^5}
\]
7. **Now we need to divide by \( (3^6)^{12} \)**:
\[
(3^6)^{12} = 3^{6 \cdot 12} = 3^{72}
\]
8. **Now we have**:
\[
\frac{2^{75} \cdot 3^3}{(2^{11} - 13)^5 \cdot 3^{72}} = \frac{2^{75}}{(2^{11} - 13)^5} \cdot \frac{3^3}{3^{72}} = \frac{2^{75}}{(2^{11} - 13)^5} \cdot 3^{3 - 72} = \frac{2^{75}}{(2^{11} - 13)^5} \cdot 3^{-69}
\]
9. **Final expression**:
\[
\frac{2^{75}}{(2^{11} - 13)^5 \cdot 3^{69}}
\]
Thus, the final result is:
\[
\frac{2^{75}}{(2^{11} - 13)^5 \cdot 3^{69}}
\]
Quick Answer
\[
\frac{2^{75}}{(2^{11} - 13)^5 \cdot 3^{69}}
\]
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