$$ (72\cdot 78\cdot 85\cdot 90\cdot 98)\div (68\cdot 84\cdot 91\cdot 108). $$

To solve the expression

\[
\frac{72 \cdot 78 \cdot 85 \cdot 90 \cdot 98}{68 \cdot 84 \cdot 91 \cdot 108},
\]

we can approach this by simplifying the fractions step by step. Let's start by factoring the numbers where possible.

### Factoring the Numerator:
- $72 = 2^3 \cdot 3^2$
- $78 = 2 \cdot 3 \cdot 13$
- $85 = 5 \cdot 17$
- $90 = 2 \cdot 3^2 \cdot 5$
- $98 = 2 \cdot 7^2$

### Factoring the Denominator:
- $68 = 2^2 \cdot 17$
- $84 = 2^2 \cdot 3 \cdot 7$
- $91 = 7 \cdot 13$
- $108 = 2^2 \cdot 3^3$

### Combine the Factors:
Now let's express the fraction using these factorizations:

\[
\frac{(2^3 \cdot 3^2) \cdot (2 \cdot 3 \cdot 13) \cdot (5 \cdot 17) \cdot (2 \cdot 3^2 \cdot 5) \cdot (2 \cdot 7^2)}{(2^2 \cdot 17) \cdot (2^2 \cdot 3 \cdot 7) \cdot (7 \cdot 13) \cdot (2^2 \cdot 3^3)}
\]

### Simplify:
- The factors of 2: $2^3 \times 2 \times 2 \times 2 = 2^7$ in the numerator, and $2^2 \times 2^2 \times 2^2 = 2^6$ in the denominator.
- The factors of 3: $3^2 \times 3 \times 3^2 = 3^5$ in the numerator, and $3 \times 3^3 = 3^4$ in the denominator.
- The factors of 5: $5 \times 5 = 5^2$ in the numerator, and none in the denominator.
- The factors of 7: $7^2$ in the numerator, and $7 \times 7 = 7^2$ in the denominator.
- The factors of 13: $13$ in both numerator and denominator.
- The factors of 17: $17$ in both numerator and denominator.

Simplifying:

\[
\frac{2^{7-6} \cdot 3^{5-4} \cdot 5^2 \cdot 7^{2-2}}{1} = 2^1 \cdot 3^1 \cdot 5^2 = 2 \cdot 3 \cdot 25 = 150
\]

So, the value of the expression is:

\[
\boxed{150}
\]

Would you like further details or have any questions? 

Here are some related questions for further exploration:

1. How do you factorize numbers?
2. What is the significance of the greatest common divisor (GCD) in simplifying fractions?
3. How would the problem change if one of the numbers was negative?
4. What are the properties of exponents that were used in this problem?
5. Can the order of multiplication or division affect the result of this problem?
6. How can prime factorization help in solving complex fractions?
7. How does cancellation work in simplifying complex fractions?
8. What is the role of the Least Common Multiple (LCM) in addition and subtraction of fractions?

**Tip:** Always factorize numbers first to simplify complex expressions efficiently.

Learn how to simplify the complex fraction (72 * 78 * 85 * 90 * 98) / (68 * 84 * 91 * 108) step-by-step. This problem involves factorization and fraction simplification techniques. Suitable for advanced high school students.

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