7. Write \( \frac{36 m^{4} n^{3}}{8 m} \div \frac{n^{2} m^{2}}{4 m n^{3}} \) in its simplest form and substitute the values to evaluate, given that \( m=3 \) and \( n=\frac{1}{3} \)

To simplify and evaluate the expression: \[ \frac{36 m^{4} n^{3}}{8 m} \div \frac{n^{2} m^{2}}{4 m n^{3}} \] follow these steps: ### Step 1: Rewrite the Division as Multiplication Recall that dividing by a fraction is equivalent to multiplying by its reciprocal: \[ \frac{36 m^{4} n^{3}}{8 m} \div \frac{n^{2} m^{2}}{4 m n^{3}} = \frac{36 m^{4} n^{3}}{8 m} \times \frac{4 m n^{3}}{n^{2} m^{2}} \] ### Step 2: Simplify the Coefficients Multiply the numerical coefficients: \[ \frac{36}{8} \times 4 = \frac{36 \times 4}{8} = \frac{144}{8} = 18 \] ### Step 3: Simplify the Variables Using Exponent Rules Combine like terms by adding/subtracting exponents: - **For \( m \):** \[ m^{4} \times m = m^{4+1} = m^{5} \quad \text{and} \quad m^{5} \div m^{2} = m^{5-2} = m^{3} \] - **For \( n \):** \[ n^{3} \times n^{3} = n^{3+3} = n^{6} \quad \text{and} \quad n^{6} \div n^{2} = n^{6-2} = n^{4} \] ### Step 4: Combine the Simplified Parts Putting it all together: \[ 18 m^{3} n^{4} \] ### Step 5: Substitute the Given Values (\( m = 3 \) and \( n = \frac{1}{3} \)) and Evaluate Substitute \( m = 3 \) and \( n = \frac{1}{3} \) into the simplified expression: \[ 18 m^{3} n^{4} = 18 \times 3^{3} \times \left(\frac{1}{3}\right)^{4} \] Calculate each part: - \( 3^{3} = 27 \) - \( \left(\frac{1}{3}\right)^{4} = \frac{1}{81} \) Now multiply them together: \[ 18 \times 27 \times \frac{1}{81} = 18 \times \frac{27}{81} = 18 \times \frac{1}{3} = 6 \] **Final Answer:** After simplifying, the expression is \( 18m^{2}n^{4} \). Substituting \( m = 3 \) and \( n = \frac{1}{3} \) gives a value of 2.