Simplify ((a^2a^3)^2*a^4)÷(((a^-1)^-2)^3)

${(a^{2} a^{3})}^{2} \cdot a^{4} \div {({(a^{- 1})}^{- 2})}^{3}$

Rewrite the division as a fraction.

$\frac{{(a^{2} a^{3})}^{2} \cdot a^{4}}{{({(a^{- 1})}^{- 2})}^{3}}$

Multiply $a^{2}$ by $a^{3}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(a^{2 + 3})}^{2} \cdot a^{4}}{{({(a^{- 1})}^{- 2})}^{3}}$

Add $2$ and $3$ .

$\frac{{(a^{5})}^{2} \cdot a^{4}}{{({(a^{- 1})}^{- 2})}^{3}}$

Multiply the exponents in ${(a^{5})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{a^{5 \cdot 2} a^{4}}{{({(a^{- 1})}^{- 2})}^{3}}$

Multiply $5$ by $2$ .

$\frac{a^{10} a^{4}}{{({(a^{- 1})}^{- 2})}^{3}}$

Simplify the denominator.

Multiply the exponents in ${({(a^{- 1})}^{- 2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{a^{10} a^{4}}{{(a^{- 1})}^{- 2 \cdot 3}}$

Multiply $- 2$ by $3$ .

$\frac{a^{10} a^{4}}{{(a^{- 1})}^{- 6}}$

Multiply the exponents in ${(a^{- 1})}^{- 6}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{a^{10} a^{4}}{a^{- - 6}}$

Multiply $- 1$ by $- 6$ .

$\frac{a^{10} a^{4}}{a^{6}}$

Multiply $a^{10}$ by $a^{4}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{a^{10 + 4}}{a^{6}}$

Add $10$ and $4$ .

$\frac{a^{14}}{a^{6}}$

Cancel the common factor of $a^{14}$ and $a^{6}$ .

Factor $a^{6}$ out of $a^{14}$ .

$\frac{a^{6} a^{8}}{a^{6}}$

Cancel the common factors.

Multiply by $1$ .

$\frac{a^{6} a^{8}}{a^{6} \cdot 1}$

Cancel the common factor.

$\frac{a^{6} a^{8}}{a^{6} \cdot 1}$

Rewrite the expression.

$\frac{a^{8}}{1}$

Divide $a^{8}$ by $1$ .

$a^{8}$