Questions: Quotient rule with negative exponents: Problem type 2 Simplify. 8 a^3 b^-5 c^-2 / 2 a^-8 c^6 Write your answer using only positive exponents.

Solution Steps

To simplify the given expression using the quotient rule with negative exponents, follow these steps:

1. Divide the coefficients (numerical values) directly.
2. Apply the quotient rule for exponents: subtract the exponent of the denominator from the exponent of the numerator for each variable.
3. Convert any negative exponents to positive by taking the reciprocal of the base with the negative exponent.

The coefficients in the expression are \(8\) and \(2\). Dividing these gives: \[ \frac{8}{2} = 4 \]

For the variable \(a\): \[ a^{3 - (-8)} = a^{3 + 8} = a^{11} \]

For the variable \(b\): \[ b^{-5 - 0} = b^{-5} \]

For the variable \(c\): \[ c^{-2 - 6} = c^{-2 - 6} = c^{-8} \]

Combining the results from the previous steps, we have: \[ \frac{8 a^{3} b^{-5} c^{-2}}{2 a^{-8} c^{6}} = \frac{4 a^{11}}{b^{5} c^{8}} \]

The expression can be rewritten to eliminate negative exponents: \[ \frac{4 a^{11}}{b^{5} c^{8}} = 4 a^{11} b^{-5} c^{-8} \]

Final Answer