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Sagot :
Certainly! Let's simplify each exponential expression step-by-step using the properties of exponents.
### Expression 1:
[tex]\[ \frac{\left(2 \cdot 3^{-2}\right)^3 \left(5 \cdot 3^2\right)^2}{\left(3^{-2}\right) \left(5 \cdot 2\right)^2} \][/tex]
1. Simplify inside the parentheses:
[tex]\[ (2 \cdot 3^{-2}) = 2 \cdot \frac{1}{3^2} = 2 \cdot \frac{1}{9} = \frac{2}{9} \][/tex]
[tex]\[ (5 \cdot 3^2) = 5 \cdot 9 = 45 \][/tex]
[tex]\[ (5 \cdot 2)^2 = 10^2 = 100 \][/tex]
2. Raise to the given powers:
[tex]\[ \left(\frac{2}{9}\right)^3 = \frac{2^3}{9^3} = \frac{8}{729} \][/tex]
[tex]\[ 45^2 = 2025 \][/tex]
3. Substitute back in:
[tex]\[ \frac{\frac{8}{729} \cdot 2025}{\frac{1}{9} \cdot 100} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{8 \cdot 2025}{729 \cdot 100} = \frac{8 \cdot 2025}{72900} = \frac{16200}{72900} = \frac{2}{9} \][/tex]
Final simplified expression:
[tex]\[ 2 \][/tex]
### Expression 2:
[tex]\[ (3^3)(4^0)^2(3 \cdot 2)^{-3}(2^2) \][/tex]
1. Simplify inside the parentheses:
[tex]\[ 4^0 = 1 \][/tex]
[tex]\[ (3 \cdot 2)^{-3} = 6^{-3} = \frac{1}{6^3} = \frac{1}{216} \][/tex]
2. Combine the terms:
[tex]\[ (3^3) \cdot (1) \cdot \frac{1}{216} \cdot 4 \][/tex]
3. Simplify:
[tex]\[ \frac{27 \cdot 4}{216} = \frac{108}{216} = \frac{1}{2} \][/tex]
### Expression 3:
[tex]\[ \frac{(3^7 \cdot 4^7)(2 \cdot 5)^{-3}6^2}{12^7 \cdot 5^{-1} \cdot 2^{-4}} \][/tex]
1. Simplify inside the parentheses:
[tex]\[ (2 \cdot 5)^{-3} = \frac{1}{10^3} = \frac{1}{1000} \][/tex]
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 12^7 = (2^2 \cdot 3)^7 = 2^{14} \cdot 3^7 \][/tex]
2. Combine the terms:
[tex]\[ \frac{(3^7 \cdot 4^7) \cdot \frac{1}{1000} \cdot 36}{2^{14} \cdot 3^7 \cdot \frac{1}{5}} \][/tex]
3. Simplify:
[tex]\[ \frac{(3^7 \cdot 4^7) \cdot 36}{2^{14} \cdot 3^7 \cdot 1000 \cdot 5} = \frac{4^72 \cdot 50}{2^{7+7}} = \frac{4^7 \cdot 72}{3^5 \cdot 3^{10}} = \frac{3^{5} \cdots 7}{2^{10}} \][/tex]
### Expression 4:
[tex]\[ \frac{(2 \cdot 3)^{-1} \cdot 2^0}{(2 \cdot 3)^{-1}} \][/tex]
1. Simplify inside the parentheses:
[tex]\[ 2^0 = 1 \][/tex]
[tex]\[ \frac{(2 \cdot 3)^{-1} \cdot 1}{(2 \cdot 3)^{-1}} = 1 \][/tex]
### Final Answers:
- Expression 1 simplifies to [tex]\( 2 \)[/tex]
- Expression 2 simplifies to [tex]\( \frac{1}{2} \)[/tex]
- Expression 3 simplifies to [tex]\( 2.88 \)[/tex]
- Expression 4 simplifies to [tex]\( 1 \)[/tex]
### Expression 1:
[tex]\[ \frac{\left(2 \cdot 3^{-2}\right)^3 \left(5 \cdot 3^2\right)^2}{\left(3^{-2}\right) \left(5 \cdot 2\right)^2} \][/tex]
1. Simplify inside the parentheses:
[tex]\[ (2 \cdot 3^{-2}) = 2 \cdot \frac{1}{3^2} = 2 \cdot \frac{1}{9} = \frac{2}{9} \][/tex]
[tex]\[ (5 \cdot 3^2) = 5 \cdot 9 = 45 \][/tex]
[tex]\[ (5 \cdot 2)^2 = 10^2 = 100 \][/tex]
2. Raise to the given powers:
[tex]\[ \left(\frac{2}{9}\right)^3 = \frac{2^3}{9^3} = \frac{8}{729} \][/tex]
[tex]\[ 45^2 = 2025 \][/tex]
3. Substitute back in:
[tex]\[ \frac{\frac{8}{729} \cdot 2025}{\frac{1}{9} \cdot 100} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{8 \cdot 2025}{729 \cdot 100} = \frac{8 \cdot 2025}{72900} = \frac{16200}{72900} = \frac{2}{9} \][/tex]
Final simplified expression:
[tex]\[ 2 \][/tex]
### Expression 2:
[tex]\[ (3^3)(4^0)^2(3 \cdot 2)^{-3}(2^2) \][/tex]
1. Simplify inside the parentheses:
[tex]\[ 4^0 = 1 \][/tex]
[tex]\[ (3 \cdot 2)^{-3} = 6^{-3} = \frac{1}{6^3} = \frac{1}{216} \][/tex]
2. Combine the terms:
[tex]\[ (3^3) \cdot (1) \cdot \frac{1}{216} \cdot 4 \][/tex]
3. Simplify:
[tex]\[ \frac{27 \cdot 4}{216} = \frac{108}{216} = \frac{1}{2} \][/tex]
### Expression 3:
[tex]\[ \frac{(3^7 \cdot 4^7)(2 \cdot 5)^{-3}6^2}{12^7 \cdot 5^{-1} \cdot 2^{-4}} \][/tex]
1. Simplify inside the parentheses:
[tex]\[ (2 \cdot 5)^{-3} = \frac{1}{10^3} = \frac{1}{1000} \][/tex]
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 12^7 = (2^2 \cdot 3)^7 = 2^{14} \cdot 3^7 \][/tex]
2. Combine the terms:
[tex]\[ \frac{(3^7 \cdot 4^7) \cdot \frac{1}{1000} \cdot 36}{2^{14} \cdot 3^7 \cdot \frac{1}{5}} \][/tex]
3. Simplify:
[tex]\[ \frac{(3^7 \cdot 4^7) \cdot 36}{2^{14} \cdot 3^7 \cdot 1000 \cdot 5} = \frac{4^72 \cdot 50}{2^{7+7}} = \frac{4^7 \cdot 72}{3^5 \cdot 3^{10}} = \frac{3^{5} \cdots 7}{2^{10}} \][/tex]
### Expression 4:
[tex]\[ \frac{(2 \cdot 3)^{-1} \cdot 2^0}{(2 \cdot 3)^{-1}} \][/tex]
1. Simplify inside the parentheses:
[tex]\[ 2^0 = 1 \][/tex]
[tex]\[ \frac{(2 \cdot 3)^{-1} \cdot 1}{(2 \cdot 3)^{-1}} = 1 \][/tex]
### Final Answers:
- Expression 1 simplifies to [tex]\( 2 \)[/tex]
- Expression 2 simplifies to [tex]\( \frac{1}{2} \)[/tex]
- Expression 3 simplifies to [tex]\( 2.88 \)[/tex]
- Expression 4 simplifies to [tex]\( 1 \)[/tex]
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