To simplify the expression

$r^{4} \cdot \frac{7r^{-5}}{12} \cdot 7r^{-7}$

we’ll work through each part step-by-step.

Step 1: Simplify the Expression

Rewrite the expression by consolidating the coefficients and powers of $r$ separately:

$r^{4} \cdot 7r^{-5} \cdot \frac{7r^{-7}}{12} = \frac{7 \cdot 7 \cdot r^{4} \cdot r^{-5} \cdot r^{-7}}{12}$

Step 2: Multiply the Coefficients

Multiply $7 \cdot 7$ to get 49:

$\frac{49 \cdot r^{4} \cdot r^{-5} \cdot r^{-7}}{12}$

Step 3: Combine the Exponents of $r$

Use the rule $r^a \cdot r^b = r^{a+b}$:

$r^{4} \cdot r^{-5} \cdot r^{-7} = r^{4 - 5 - 7} = r^{-8}$

Step 4: Rewrite with Positive Exponents

Now, we have:

$\frac{49 \cdot r^{-8}}{12} = \frac{49}{12} \cdot \frac{1}{r^8} = \frac{49}{12r^8}$

Final Answer

The simplified expression is:

$\frac{49}{12r^8}$

Would you like more details or have any questions?

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Here are some related questions:

1. How would the answer change if variables were not assumed to be positive?
2. Can you explain why we need to keep exponents positive in the final answer?
3. What happens if we add or subtract two fractions with $r$ terms?
4. How does this process change if the base of the exponents is a different variable?
5. What rules are used to combine exponents when multiplying?

Tip: Remember, when simplifying, keep terms with the same base together to make exponent operations easier.