Simplify. Express your answer using positive exponents. \( \frac{4 g^{8} h}{4 g^{7} h^{3} \cdot g^{4} h^{6}} \) \( \square \) Submit

We start with the given expression: \[ \frac{4 g^{8} h}{4 g^{7} h^{3} \cdot g^{4} h^{6}} \] **Step 1: Simplify the Denominator** Multiply the factors in the denominator: \[ 4 g^{7} h^{3} \cdot g^{4} h^{6} = 4 g^{7+4} h^{3+6} = 4 g^{11} h^{9} \] **Step 2: Write the Fraction as a Single Fraction** Now the expression becomes: \[ \frac{4 g^8 h}{4 g^{11} h^9} \] **Step 3: Cancel Common Factors** Cancel the factor of \(4\): \[ \frac{g^8 h}{g^{11} h^9} \] Subtract the exponents for like bases: \[ g^8 \div g^{11} = g^{8-11} = g^{-3} \quad \text{and} \quad h \div h^9 = h^{1-9} = h^{-8} \] **Step 4: Express Using Positive Exponents** The expression becomes: \[ g^{-3} h^{-8} = \frac{1}{g^3 h^8} \] Thus, the simplified expression is: \[ \frac{1}{g^{3} h^{8}} \]