Simplify ((x^(3s-1)y^(s+2))/(x^(1-s)y^s))^(-r)

${(\frac{x^{3 s - 1} y^{s + 2}}{x^{1 - s} y^{s}})}^{- r}$

Step 1

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x^{3 s - 1}$ and $x^{1 - s}$ .

Factor $x^{1 - s}$ out of $x^{3 s - 1} y^{s + 2}$ .

${(\frac{x^{1 - s} (x^{3 s - 1 - (1 - s)} y^{s + 2})}{x^{1 - s} y^{s}})}^{- r}$

Cancel the common factors.

Factor $x^{1 - s}$ out of $x^{1 - s} y^{s}$ .

${(\frac{x^{1 - s} (x^{3 s - 1 - (1 - s)} y^{s + 2})}{x^{1 - s} (y^{s})})}^{- r}$

Cancel the common factor.

${(\frac{x^{1 - s} (x^{3 s - 1 - (1 - s)} y^{s + 2})}{x^{1 - s} y^{s}})}^{- r}$

Rewrite the expression.

${(\frac{x^{3 s - 1 - (1 - s)} y^{s + 2}}{y^{s}})}^{- r}$

Cancel the common factor of $y^{s + 2}$ and $y^{s}$ .

Factor $y^{s}$ out of $x^{3 s - 1 - (1 - s)} y^{s + 2}$ .

${(\frac{y^{s} (x^{3 s - 1 - (1 - s)} y^{2})}{y^{s}})}^{- r}$

Cancel the common factors.

Multiply by $1$ .

${(\frac{y^{s} (x^{3 s - 1 - (1 - s)} y^{2})}{y^{s} \cdot 1})}^{- r}$

Cancel the common factor.

${(\frac{y^{s} (x^{3 s - 1 - (1 - s)} y^{2})}{y^{s} \cdot 1})}^{- r}$

Rewrite the expression.

${(\frac{x^{3 s - 1 - (1 - s)} y^{2}}{1})}^{- r}$

Divide $x^{3 s - 1 - (1 - s)} y^{2}$ by $1$ .

${(x^{3 s - 1 - (1 - s)} y^{2})}^{- r}$

Step 2

Simplify each term.

Apply the distributive property.

${(x^{3 s - 1 - 1 \cdot 1 - - s} y^{2})}^{- r}$

Multiply $- 1$ by $1$ .

${(x^{3 s - 1 - 1 - - s} y^{2})}^{- r}$

Multiply $- - s$ .

Multiply $- 1$ by $- 1$ .

${(x^{3 s - 1 - 1 + 1 s} y^{2})}^{- r}$

Multiply $s$ by $1$ .

${(x^{3 s - 1 - 1 + s} y^{2})}^{- r}$

Step 3

Simplify by adding terms.

Add $3 s$ and $s$ .

${(x^{4 s - 1 - 1} y^{2})}^{- r}$

Simplify the expression.

Subtract $1$ from $- 1$ .

${(x^{4 s - 2} y^{2})}^{- r}$

Apply the product rule to $x^{4 s - 2} y^{2}$ .

${(x^{4 s - 2})}^{- r} {(y^{2})}^{- r}$

Multiply the exponents in ${(x^{4 s - 2})}^{- r}$ .

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

$x^{(4 s - 2) (- r)} {(y^{2})}^{- r}$

Apply the distributive property.

$x^{4 s (- r) - 2 (- r)} {(y^{2})}^{- r}$

Rewrite using the commutative property of multiplication.

$x^{4 \cdot - 1 s r - 2 (- r)} {(y^{2})}^{- r}$

Multiply $- 1$ by $- 2$ .

$x^{4 \cdot - 1 s r + 2 r} {(y^{2})}^{- r}$

Multiply $4$ by $- 1$ .

$x^{- 4 s r + 2 r} {(y^{2})}^{- r}$

Multiply the exponents in ${(y^{2})}^{- r}$ .

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

$x^{- 4 s r + 2 r} y^{2 (- r)}$

Multiply $- 1$ by $2$ .

$x^{- 4 s r + 2 r} y^{- 2 r}$

Simplify ((x^(3s-1)y^(s+2))/(x^(1-s)y^s))^(-r)

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