Simplify the following expressions and write them with positive exponents.

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Here, we need to use the power rule, the quotient rule, and the product rule of exponents.

Power rule: (x^a)^b = x^(a*b)

Quotient rule: x^a / x^b = x^(a-b)

Product rule: x^a * x^b = x^(a+b)

a) x^2*x^3 = x^(2+3) = x^5

b) a^-2*a^-3 = a^(-2+ -3) = a^-5 = 1/a^5

c) a^5 / a^2 = a^(5-2) = a^3

d) a^-3 / a^7 = a^(-3-7) = a^-10 = 1/a^10

e) (x^2)^3 = x^(2*3) = x^6

f) (x^-2)^5 = x^(-2*5) = x^-10 = 1/x^10

g) (x^2*y^3)^3 = (x^2)^3 * (y^3)^3 = x^(2*3) * y^(3*3) = x^6*y^9

h) (x^-2*y^3)^8 = (x^-2)^8 * (y^3)^8 = x^(-2*8)*y^(3*8) = x^-16*y^24 = y^24 / x^16

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