Which expression is equivalent to
`\frac{40x^{-3}y^{5}}{8x^{-2}y^{-2}}`
for all values for `x` and `y` where the expression is defined?
To simplify the expression, we can divide the numerator by the denominator:
`\frac{40x^{-3}y^{5}}{8x^{-2}y^{-2}} = \frac{40}{8} \cdot \frac{x^{-3}}{x^{-2}} \cdot \frac{y^{5}}{y^{-2}}`
Simplifying further, we can divide 40 by 8 to get 5:
`\frac{40}{8} \cdot \frac{x^{-3}}{x^{-2}} \cdot \frac{y^{5}}{y^{-2}} = 5 \cdot x^{-3-(-2)} \cdot y^{5-(-2)}`
Simplifying the exponents will yield:
`5 \cdot x^{-1} \cdot y^{7}`
Multiplying 5 with x^-1 and y^7 gives the simplified expression:
`5x^{-1}y^{7}`
Therefore, the expression `\frac{40x^{-3}y^{5}}{8x^{-2}y^{-2}}` is equivalent to `5x^{-1}y^{7}`.