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 \( m \) ?

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