Simplify the following expressions, leaving your answers in positive index.

1. Simplify the power of the bracket and the zero index term.

$(-a^2)^3 = (-1)^3(a^2)^3 = -a^6$

$b^0=1$, so $4b^0=4$.

2. Divide the simplified terms and write the answer with positive indices.

$(-a^2)^3 \div 4b^0 = -a^6 \div 4 = -\frac{a^6}{4}$

1. Expand the power of the first bracket using index laws.

$(a^{-1}b)^2=(a^{-1})^2b^2=a^{-2}b^2$

2. Rewrite the square root as a fractional index and combine powers of $b$, leaving positive indices.

$(\sqrt{b})^3=(b^{\frac12})^3=b^{\frac32}$

$a^{-2}b^2\times b^{\frac32}=a^{-2}b^{2+\frac32}=a^{-2}b^{\frac72}=\frac{b^{\frac72}}{a^2}$