Simplify (a^-5b^8c^-12)(a^7b^-3c^7)

$(a^{- 5} b^{8} c^{- 12}) (a^{7} b^{- 3} c^{7})$

Multiply $a^{- 5}$ by $a^{7}$ by adding the exponents.

Move $a^{7}$ .

$a^{7} a^{- 5} b^{8} c^{- 12} (b^{- 3} c^{7})$

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

$a^{7 - 5} b^{8} c^{- 12} (b^{- 3} c^{7})$

Subtract $5$ from $7$ .

$a^{2} b^{8} c^{- 12} (b^{- 3} c^{7})$

Multiply $b^{8}$ by $b^{- 3}$ by adding the exponents.

Move $b^{- 3}$ .

$a^{2} (b^{- 3} b^{8}) c^{- 12} c^{7}$

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

$a^{2} b^{- 3 + 8} c^{- 12} c^{7}$

Add $- 3$ and $8$ .

$a^{2} b^{5} c^{- 12} c^{7}$

Multiply $c^{- 12}$ by $c^{7}$ by adding the exponents.

Move $c^{7}$ .

$a^{2} b^{5} (c^{7} c^{- 12})$

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

$a^{2} b^{5} c^{7 - 12}$

Subtract $12$ from $7$ .

$a^{2} b^{5} c^{- 5}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$a^{2} b^{5} \frac{1}{c^{5}}$

Multiply $a^{2} b^{5} \frac{1}{c^{5}}$ .

Combine $a^{2}$ and $\frac{1}{c^{5}}$ .

$b^{5} \frac{a^{2}}{c^{5}}$

Combine $b^{5}$ and $\frac{a^{2}}{c^{5}}$ .

$\frac{b^{5} a^{2}}{c^{5}}$