The expression $$ \frac{\left(x^{4} x^{5}\right)^{3} x^{7}}{\left(x^{3} x^{-2}\right) x^{2}} $$ is equivalent to $$ x^{m} $$. What is the value of $$ m $$ ?

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To simplify the expression $$ \frac{\left(x^{4} x^{5}\right)^{3} x^{7}}{\left(x^{3} x^{-2}\right) x^{2}} $$ and express it in the form $$ x^{m} $$, follow these steps:

1. **Simplify the Numerator:**
   $$
   \left(x^{4} x^{5}\right)^{3} x^{7}
   $$
   Combine the exponents inside the parentheses:
   $$
   x^{4+5} = x^{9}
   $$
   Now raise this to the power of 3:
   $$
   \left(x^{9}\right)^{3} = x^{27}
   $$
   Multiply by $$ x^{7} $$:
   $$
   x^{27} \cdot x^{7} = x^{34}
   $$

2. **Simplify the Denominator:**
   $$
   \left(x^{3} x^{-2}\right) x^{2}
   $$
   Combine the exponents inside the parentheses:
   $$
   x^{3-2} = x^{1}
   $$
   Multiply by $$ x^{2} $$:
   $$
   x^{1} \cdot x^{2} = x^{3}
   $$

3. **Combine Numerator and Denominator:**
   $$
   \frac{x^{34}}{x^{3}} = x^{34-3} = x^{31}
   $$

Therefore, the value of $$ m $$ is **31**.

**Answer:** 31
$$ m = 31 $$

The expression \( \frac{\left(x^{4} x^{5}\right)^{3} x^{7}}{\left(x^{3} x^{-2}\right) x^{2}} \) is equivalent to \( x^{m} \). What is the value of \( m \) ?

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