4)) Simplify. Express your answer using positive exponent \( \frac{8 a^{-1} b^{-1} c^{-6}}{8 a^{7} b^{7} c^{-3}} \)

To simplify the expression \( \frac{8 a^{-1} b^{-1} c^{-6}}{8 a^{7} b^{7} c^{-3}} \), we begin by canceling the common factor of 8 in the numerator and denominator. This leaves us with: \[ \frac{a^{-1} b^{-1} c^{-6}}{a^{7} b^{7} c^{-3}} \] Next, we apply the properties of exponents. For the \( a \) terms: \[ a^{-1} \div a^{7} = a^{-1 - 7} = a^{-8} \] For the \( b \) terms: \[ b^{-1} \div b^{7} = b^{-1 - 7} = b^{-8} \] For the \( c \) terms: \[ c^{-6} \div c^{-3} = c^{-6 - (-3)} = c^{-6 + 3} = c^{-3} \] Putting it all together: \[ \frac{a^{-8} b^{-8} c^{-3}}{1} = a^{-8} b^{-8} c^{-3} \] To express the answer using positive exponents, we rewrite the expression: \[ \frac{1}{a^{8} b^{8} c^{3}} \] So, the final simplified expression is: \[ \frac{1}{a^{8} b^{8} c^{3}} \]