Simplify: `(m^8p^7r^(12))/(m^3 r^9p) xx p^2 r^3 m^4`

Answer

To simplify the expression \((m^8p^7r^{12})/(m^3 r^9p) \times p^2 r^3 m^4\), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression clearly:
\[ \frac{m^8 p^7 r^{12}}{m^3 r^9 p} \times p^2 r^3 m^4 \]

### Step 2: Combine the fractions
We can combine the multiplication and division into a single fraction:
\[ = \frac{m^8 p^7 r^{12} \cdot p^2 r^3 m^4}{m^3 r^9 p} \]

### Step 3: Simplify the numerator
Now, we simplify the numerator:
\[ = \frac{m^8 m^4 p^7 p^2 r^{12} r^3}{m^3 r^9 p} \]
Using the property \(x^a \cdot x^b = x^{a+b}\), we get:
\[ = \frac{m^{8+4} p^{7+2} r^{12+3}}{m^3 r^9 p} \]
This simplifies to:
\[ = \frac{m^{12} p^9 r^{15}}{m^3 r^9 p} \]

### Step 4: Simplify the fraction
Now we will simplify the fraction using the property \(\frac{x^a}{x^b} = x^{a-b}\):
\[ = m^{12-3} p^{9-1} r^{15-9} \]
This gives us:
\[ = m^9 p^8 r^6 \]

### Step 5: Final result
Thus, the simplified expression is:
\[ \boxed{m^9 p^8 r^6} \]

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