Respuesta :
Answer:
Sure, let's break it down step by step:
1. Start by simplifying the numerator and denominator separately.
\(3^{-2} \cdot 2^3 \cdot 5^{-3}\) in the numerator and \(3^5 \cdot 2^{-6} \cdot 5^4\) in the denominator.
\(3^{-2} = \frac{1}{3^2}\), \(5^{-3} = \frac{1}{5^3}\), \(2^{-6} = \frac{1}{2^6}\).
The expression becomes \(\frac{1}{3^2} \cdot 2^3 \cdot \frac{1}{5^3}\) in the numerator and \(3^5 \cdot \frac{1}{2^6} \cdot 5^4\) in the denominator.
2. Simplify further within the numerator and denominator.
\(\frac{1}{3^2} = \frac{1}{9}\), \(\frac{1}{5^3} = \frac{1}{125}\), \(\frac{1}{2^6} = \frac{1}{64}\).
The expression becomes \(\frac{1}{9} \cdot 2^3 \cdot \frac{1}{125}\) in the numerator and \(3^5 \cdot \frac{1}{64} \cdot 5^4\) in the denominator.
3. Calculate the powers.
\(2^3 = 8\), \(3^5 = 243\), \(5^4 = 625\).
The expression becomes \(\frac{8}{9 \cdot 125}\) in the numerator and \(\frac{243}{64 \cdot 625}\) in the denominator.
4. Simplify further.
\(\frac{8}{9 \cdot 125} = \frac{8}{1125}\), and \(\frac{243}{64 \cdot 625} = \frac{243}{40000}\).
5. Divide the numerator by the denominator.
\(\frac{8}{1125} \div \frac{243}{40000} = \frac{8}{1125} \cdot \frac{40000}{243}\).
6. Multiply the numerators and denominators.
\(\frac{8 \cdot 40000}{1125 \cdot 243}\).
7. Simplify the fraction.
\(\frac{320000}{273375}\).
So, \(\frac{3^{-2} \cdot 2^3 \cdot 5^{-3}}{3^5 \cdot 2^{-6} \cdot 5^4} = \frac{320000}{273375}\).
