Ения: (a ^ 2 + ax)/x / ((ax)/(x ^ 2)) 7a + (2a - 7e ^ 2)/6 12/(2a - a ^ 1) - 6/a (xy + y ^ 2)/(15x) * (3x)/(x + y) 6/(x + 3) - x/((x + 3) ^ 2) / (x/(x ^ 2 - 9)) - (2x + 18)/(x + 3)

To simplify these expressions, we'll follow the order of operations (PEMDAS/BODMAS).

Expression 1: (a^2 + ax)/x / ((ax)/(x^2))

To simplify this expression, we can multiply the numerator and denominator by x to eliminate the division.

(a^2 + ax) * x / (ax) * (x^2/x) = a^2x + ax^2 / ax^2 = (a^2 + ax) / a = a(a + x) / a = a + x

Final answer: a + x

Expression 2: 7a + (2a - 7e^2)/6

This expression is already simplified.

Final answer: 7a + (2a - 7e^2)/6

Expression 3: 12/(2a - a^1) - 6/a

Simplify the denominator first: 2a - a^1 = 2a - a = a

Then, rewrite the expression:

12/a - 6/a = (12 - 6)/a = 6/a

Final answer: 6/a

Expression 4: (xy + y^2)/(15x) * (3x)/(x + y)

Multiply the numerators together, and the denominators together:

(xy + y^2) * (3x) / (15x) * (x + y) = (3xy^2 + 3x^2y) / (15x^2 + 15xy)

Factor out 3xy from the numerator:

3xy(y + x) / 15x(y + x)

Cancel out common terms:

3y / 15 = y / 5

Final answer: y/5

Expression 5: 6/(x + 3) - x/((x + 3)^2) / (x/(x^2 - 9)) - (2x + 18)/(x + 3)

First, simplify the expression in the denominator:

x + 3 = (x + 3)(x - 3)

Now rewrite the expression:

6/(x + 3) - x/((x + 3)^2) / (x/(x^2 - 9)) - (2x + 18)/(x + 3) = 6/(x + 3) - x/(x + 3)(x - 3) / (x/(x + 3)(x - 3)) - (2x + 18)/(x + 3)

Now look for common denominators and combine like terms:

= 6 - x/(x - 3) - 1 - (2(x + 9))/(x + 3) = 5 - x/(x - 3) - 2(x + 9)/(x + 3) = 5 - x/(x - 3) - (2x + 18)/(x + 3)

Combine the terms over common denominators:

= (5(x - 3) - x - (2x + 18))/(x - 3)(x + 3) = (5x - 15 - x - 2x - 18)/(x - 3)(x + 3) = (2x - 33)/(x - 3)(x + 3)

Final answer: (2x - 33)/(x - 3)(x + 3)

Remember, always double-check your work when simplifying expressions to ensure accuracy.