Use the properties of exponents to rewrite each expression so that it contains only positive exponents and simplify.
(x^-1y^-2/x^-2y^3)^2
[(1/xy^2)/(y^3/x^2)]^2
[x/y^5]^2
x^2/y^10
To rewrite the expression and simplify it, we can use the properties of exponents.
Step 1: Bring the numerator and denominator of the fraction to their respective exponents.
(x^-1y^-2/x^-2y^3)^2 becomes ((x^-1)^2(y^-2)^2/(x^-2)^2(y^3)^2).
Step 2: Apply the power of a power rule, which states that raising an exponent to another exponent multiplies the exponents.
((x^-1)^2(y^-2)^2/(x^-2)^2(y^3)^2) becomes (x^(-1*2)y^(-2*2)/x^(-2*2)y^(3*2)).
Simplifying further:
(x^(-2)y^(-4)/x^(-4)y^6).
Step 3: Use the quotient of powers property, which states that to divide two exponential expressions with the same base, we subtract the exponents.
(x^(-2)y^(-4)/x^(-4)y^6) become x^(-2-(-4)) y^(-4-6).
Simplifying:
(x^2y^(-10)).
Step 4: Finally, rewrite the expression with only positive exponents by applying the negative exponent rule, which states that x^-n is equal to 1/x^n.
(x^2y^(-10)) becomes (x^2/y^10).
So, the expression (x^-1y^-2/x^-2y^3)^2, when rewritten with only positive exponents and simplified, is equal to (x^2/y^10).