Simplify the exponential expression (-35a^2*b^3 / (7a^8*b^-5))^3.

Step-by-step solution

1. 1

   Simplify inside the bracket before applying the outer power. Cubing first would triple the amount of arithmetic, so reduce the fraction while the exponents are small. Start with the coefficients: $\frac{-35}{7}=-5$.

2. 2

   Apply the quotient rule to each variable. Subtracting exponents gives $\frac{a^{2}}{a^{8}}=a^{2-8}=a^{-6},\qquad \frac{b^{3}}{b^{-5}}=b^{3-(-5)}=b^{8}.$ The second one is the step people get wrong: subtracting a negative exponent adds. So the bracket equals $-5a^{-6}b^{8}$.

3. 3

   Raise every factor to the third power. $\left(-5a^{-6}b^{8}\right)^{3}=(-5)^{3}\left(a^{-6}\right)^{3}\left(b^{8}\right)^{3}=-125\,a^{-18}b^{24}.$ Note $(-5)^{3}=-125$ is negative because the exponent is odd.

4. 4

   Rewrite with positive exponents. Since $a^{-18}=\frac{1}{a^{18}}$, $-125a^{-18}b^{24}=-\frac{125b^{24}}{a^{18}} .$

5. 5

   Check with a numerical substitution. Taking $a=b=2$: the bracket is $\frac{-35\cdot4\cdot8}{7\cdot256\cdot 2^{-5}}=\frac{-1120}{56}=-20$, and $(-20)^{3}=-8000$; the answer gives $-\frac{125\cdot 2^{24}}{2^{18}}=-125\cdot 2^{6}=-8000$, which matches.