Simplify. \[ 2 v^{-6} \cdot 9 w u^{-6} \cdot 5 u^{6} v^{9} w^{-8} \] Use only positive exponents in your answer.
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To simplify the expression \( 2 v^{-6} \cdot 9 w u^{-6} \cdot 5 u^{6} v^{9} w^{-8} \), we start by multiplying the coefficients and combining the variables. First, multiply the coefficients: \[ 2 \cdot 9 \cdot 5 = 90. \] Next, combine the variables: - For \( v \): \[ v^{-6} \cdot v^{9} = v^{-6 + 9} = v^{3}. \] - For \( w \): \[ w \cdot w^{-8} = w^{1 - 8} = w^{-7}. \] - For \( u \): \[ u^{-6} \cdot u^{6} = u^{-6 + 6} = u^{0} = 1. \] Now we can write the simplified expression: \[ 90 v^{3} w^{-7}. \] To express everything with positive exponents, we can rewrite \( w^{-7} \) as \( \frac{1}{w^{7}} \). Therefore, the expression becomes: \[ \frac{90 v^{3}}{w^{7}}. \] Thus, the final answer is: \[ \frac{90 v^{3}}{w^{7}}. \]
