Evaluate (-xy)^2*8x^-7y^-2

${(- x y)}^{2} \cdot 8 x^{- 7} y^{- 2}$

Step 1

Rewrite using the commutative property of multiplication.

$8 {(- x y)}^{2} x^{- 7} y^{- 2}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- x y$ .

$8 ({(- x)}^{2} y^{2}) x^{- 7} y^{- 2}$

Apply the product rule to $- x$ .

$8 ({(- 1)}^{2} x^{2} y^{2}) x^{- 7} y^{- 2}$

Step 3

Multiply $x^{2}$ by $x^{- 7}$ by adding the exponents.

Move $x^{- 7}$ .

$8 ({(- 1)}^{2} (x^{- 7} x^{2}) y^{2}) y^{- 2}$

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

$8 ({(- 1)}^{2} x^{- 7 + 2} y^{2}) y^{- 2}$

Add $- 7$ and $2$ .

$8 ({(- 1)}^{2} x^{- 5} y^{2}) y^{- 2}$

Step 4

Multiply $y^{2}$ by $y^{- 2}$ by adding the exponents.

Move $y^{- 2}$ .

$8 ({(- 1)}^{2} x^{- 5} (y^{- 2} y^{2}))$

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

$8 ({(- 1)}^{2} x^{- 5} y^{- 2 + 2})$

Add $- 2$ and $2$ .

$8 ({(- 1)}^{2} x^{- 5} y^{0})$

Step 5

Simplify $8 ({(- 1)}^{2} x^{- 5} y^{0})$ .

$8 ({(- 1)}^{2} x^{- 5})$

Step 6

Raise $- 1$ to the power of $2$ .

$8 (1 x^{- 5})$

Step 7

Multiply $x^{- 5}$ by $1$ .

$8 x^{- 5}$

Step 8

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

$8 \frac{1}{x^{5}}$

Step 9

Combine $8$ and $\frac{1}{x^{5}}$ .

$\frac{8}{x^{5}}$