Simplify (x^(-u)y^v)^(-a)(x^(-2u)y^(3v))^a

${(x^{- u} y^{v})}^{- a} {(x^{- 2 u} y^{3 v})}^{a}$

Apply the product rule to $x^{- u} y^{v}$ .

${(x^{- u})}^{- a} {(y^{v})}^{- a} {(x^{- 2 u} y^{3 v})}^{a}$

Multiply the exponents in ${(x^{- u})}^{- a}$ .

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

$x^{- u (- a)} {(y^{v})}^{- a} {(x^{- 2 u} y^{3 v})}^{a}$

Rewrite using the commutative property of multiplication.

$x^{- 1 \cdot - 1 u a} {(y^{v})}^{- a} {(x^{- 2 u} y^{3 v})}^{a}$

Multiply $- 1$ by $- 1$ .

$x^{1 u a} {(y^{v})}^{- a} {(x^{- 2 u} y^{3 v})}^{a}$

Multiply $u$ by $1$ .

$x^{u a} {(y^{v})}^{- a} {(x^{- 2 u} y^{3 v})}^{a}$

Multiply the exponents in ${(y^{v})}^{- a}$ .

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

$x^{u a} y^{v (- a)} {(x^{- 2 u} y^{3 v})}^{a}$

Rewrite using the commutative property of multiplication.

$x^{u a} y^{- v a} {(x^{- 2 u} y^{3 v})}^{a}$

Apply the product rule to $x^{- 2 u} y^{3 v}$ .

$x^{u a} y^{- v a} ({(x^{- 2 u})}^{a} {(y^{3 v})}^{a})$

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

$x^{u a} y^{- v a} (x^{- 2 u a} {(y^{3 v})}^{a})$

Multiply $x^{u a}$ by $x^{- 2 u a}$ by adding the exponents.

Move $x^{- 2 u a}$ .

$x^{- 2 u a} x^{u a} y^{- v a} {(y^{3 v})}^{a}$

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

$x^{- 2 u a + u a} y^{- v a} {(y^{3 v})}^{a}$

Add $- 2 u a$ and $u a$ .

$x^{- u a} y^{- v a} {(y^{3 v})}^{a}$

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

$x^{- u a} y^{- v a} y^{3 v a}$

Multiply $y^{- v a}$ by $y^{3 v a}$ by adding the exponents.

Move $y^{3 v a}$ .

$x^{- u a} (y^{3 v a} y^{- v a})$

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

$x^{- u a} y^{3 v a - v a}$

Subtract $v a$ from $3 v a$ .

$x^{- u a} y^{2 v a}$