Find the Domain ((xy)^-3)/((x^-5y)^3)

Problem

((x*y)(−3))/((x(−5)*y)3)

Solution

1. Simplify the numerator by applying the power of a product rule (a*b)n=an*bn

(x(−3)*y(−3))/((x(−5)*y)3)

2. Simplify the denominator by applying the power of a product rule and the power of a power rule (am)n=a(m*n)

(x(−3)*y(−3))/(x(−15)*y3)

3. Combine the terms by using the quotient rule (am)/(an)=a(m−n)

x(−3−(−15))*y(−3−3)

4. Simplify the exponents to find the reduced expression.

x12*y(−6)

5. Rewrite the expression with positive exponents to identify potential restrictions.

(x12)/(y6)

6. Identify the restrictions on the variables. In the original expression, x and y appear in the denominator or with negative exponents. For the expression to be defined, the bases of negative powers must be non-zero.

x≠0

y≠0

Final Answer

Domain={[(x,y)∈ℝ2,x≠0,y≠0]}

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