Simplify. Assume all variables are positive.
r
4

7
r–
5

7

r–
12

7
Write your answer in the form A or 
A

B
, where A and B are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.


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?

---

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.

Simplify. Assume all variables are positive. r^4 * (7r^-5 / 12) * 7r^-7. Write your answer in the form A or A / B, where A and B are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

Discover how to simplify the expression r^4 * (7r^-5 / 12) * 7r^-7 step-by-step. This problem involves key algebra concepts, such as exponent rules and fraction simplification. Ideal for high school students in Grades 9-10.

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