Simplify the exponential expression: the quantity negative 35 a squared b cubed over 7 a to the eighth b to the negative fifth, all cubed.

Step-by-step solution

1. 1

   Simplify inside the parentheses first. Cubing a messy quotient triples every mistake, so reduce the base before applying the outer power.

2. 2

   Divide the coefficients. $-35 \div 7 = -5$.

3. 3

   Apply the quotient rule to each variable. For $a$: $a^{2-8} = a^{-6}$. For $b$: $b^{3-(-5)} = b^{8}$. Subtracting a negative exponent adds, which is where sign errors usually creep in.

4. 4

   Rewrite the base. The parentheses now hold $-5a^{-6}b^{8} = \dfrac{-5b^8}{a^6}$.

5. 5

   Raise everything to the third power. $(-5)^3 = -125$, $(b^8)^3 = b^{24}$ and $(a^6)^3 = a^{18}$; the odd power keeps the result negative.

6. 6

   State the answer with positive exponents. $\dfrac{-125b^{24}}{a^{18}}$.