To simplify this complex expression, let's go through it step-by-step. The expression is:

$\left( \frac{(2 \cdot 5)^3 \cdot (2 \cdot 5)^4 \cdot (3)^{-4}}{(5)^{-1} \cdot (5)^{-5} \cdot (5)^{-5}} \right)^{-5} \times \left( \frac{(5)^0 \cdot (2)^{-5} \cdot (3)^{-2} \cdot (3)^{-2}}{(2)^{-4} \cdot (5)^{-3} \cdot (5)^{-3}} \right)^3$

Let's simplify each part individually:

1. Simplify each term inside the parentheses.

For the first expression:

• The numerator is $(2 \cdot 5)^3 \cdot (2 \cdot 5)^4 \cdot (3)^{-4}$.

  • Using the property $(a \cdot b)^n = a^n \cdot b^n$, we can write $(2 \cdot 5)^3 = 2^3 \cdot 5^3$ and $(2 \cdot 5)^4 = 2^4 \cdot 5^4$.
  • This gives $2^3 \cdot 5^3 \cdot 2^4 \cdot 5^4 \cdot 3^{-4} = 2^{3+4} \cdot 5^{3+4} \cdot 3^{-4} = 2^7 \cdot 5^7 \cdot 3^{-4}$.

• The denominator is $5^{-1} \cdot 5^{-5} \cdot 5^{-5} = 5^{-1-5-5} = 5^{-11}$.

• So, the entire fraction becomes: $\frac{2^7 \cdot 5^7 \cdot 3^{-4}}{5^{-11}} = 2^7 \cdot 5^{7 + 11} \cdot 3^{-4} = 2^7 \cdot 5^{18} \cdot 3^{-4}$

• Raising this to the power of $-5$ gives: $\left(2^7 \cdot 5^{18} \cdot 3^{-4}\right)^{-5} = 2^{-35} \cdot 5^{-90} \cdot 3^{20}$

For the second expression:

• The numerator is $(5)^0 \cdot (2)^{-5} \cdot (3)^{-2} \cdot (3)^{-2}$.

  • Simplify to $5^0 \cdot 2^{-5} \cdot 3^{-2-2} = 2^{-5} \cdot 3^{-4}$.

• The denominator is $(2)^{-4} \cdot (5)^{-3} \cdot (5)^{-3} = 2^{-4} \cdot 5^{-6}$.

• So, the entire fraction becomes: $\frac{2^{-5} \cdot 3^{-4}}{2^{-4} \cdot 5^{-6}} = 2^{-5+4} \cdot 3^{-4} \cdot 5^{6} = 2^{-1} \cdot 3^{-4} \cdot 5^{6}$

• Raising this to the power of $3$ gives: $\left(2^{-1} \cdot 3^{-4} \cdot 5^{6}\right)^3 = 2^{-3} \cdot 3^{-12} \cdot 5^{18}$

2. Combine both parts:

The expression now becomes: $2^{-35} \cdot 5^{-90} \cdot 3^{20} \times 2^{-3} \cdot 3^{-12} \cdot 5^{18}$

Combine the exponents for each base:

• For $2$: $-35 - 3 = -38$, so we have $2^{-38}$.
• For $5$: $-90 + 18 = -72$, so we have $5^{-72}$.
• For $3$: $20 - 12 = 8$, so we have $3^8$.

3. Final simplified form:

$2^{-38} \cdot 5^{-72} \cdot 3^8 = \frac{3^8}{2^{38} \cdot 5^{72}}$

This is the fully simplified expression. Let me know if you'd like further clarification on any step.

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Related Questions

1. How do exponent rules apply to expressions with multiple bases?
2. Can this expression be simplified further if we evaluate the constants?
3. What is the significance of negative exponents in these expressions?
4. How do we handle fractions when simplifying expressions with exponents?
5. Can this technique be applied to logarithmic expressions as well?

Tip

Always keep track of each base separately when applying exponent rules to avoid calculation errors.