(a ^ 2 + ax)/x / ((ax)/(x ^ 2)) 7a + (2a - 7e ^ 2)/6 12/(2a - a ^ 2) - 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 the given expressions, let's go through each one step by step:

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

Here, we have a fraction divided by another fraction. We can simplify this by taking the reciprocal of the second fraction and then multiplying the two fractions together.

Reciprocal of ((ax)/(x^2)) is (x^2)/(ax). Therefore, the expression becomes: (a^2 + ax)/x * (x^2)/(ax)

Now, we can cancel out the common terms in numerator and denominator. The expression simplifies to: (a^2 * x^2) / (x * ax) Simplifying further, we get: ax / (ax) Finally, cancel out the common "ax" terms to get: 1

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

This expression cannot be simplified further as it has no like terms that can be combined.

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

To simplify this expression, let's factor the denominator of the first term. The expression becomes: 12 / (a(2 - a)) - 6/a

To combine the two fractions, let's find the common denominator, which is "a(2 - a)". The expression becomes: (12 - 6(2 - a))/(a(2 - a))

Expanding the numerator, we have: (12 - 12 + 6a)/(a(2 - a)) Simplifying, we get: (6a)/(a(2 - a))

Now, we can cancel out the common "a" terms. The expression simplifies to: 6 / (2 - a)

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

To simplify this expression, let's simplify each fraction separately, and then multiply them together.

The first fraction simplifies to: y(x + y)/(15x)

The second fraction remains as it is: (3x)/(x + y)

Now, we can multiply the two fractions together: (y(x + y)/(15x)) * (3x)/(x + y)

The common terms "x + y" can be canceled out, leaving us with: (3y)/(15)

Further simplifying, we get: y/5

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

In this expression, we have multiple terms and fractions. Let's simplify each individually and then combine them.

First, let's simplify the fractions: 6/(x + 3) - x/((x + 3)^2) becomes (6(x + 3) - x(x + 3))/(x + 3)^2

Further simplifying, we get: (6x + 18 - x^2 - 3x)/(x + 3)^2

Now, let's simplify the other fraction: x/(x^2 - 9) becomes x/((x + 3)(x - 3))

Finally, let's simplify the remaining term: (2x + 18)/(x + 3)

To combine all the terms, we need a common denominator. In this case, it will be (x + 3)^2(x - 3).

The expression becomes: ((6x + 18 - x^2 - 3x)(x - 3))/(x + 3)^2(x - 3) - ((x)(x + 3)(x - 3))/(x + 3)^2(x - 3) + ((2x + 18)(x + 3))/(x + 3)^2(x - 3)

Now, expand and simplify the numerator: (6x^2 - 18x + 18x - 54 - x^3 + 3x^2 - 6x + 18x^2 - 54x)/(x + 3)^2(x - 3)

Adding like terms: (5x^2 + 18x^2 - x^3 + 3x^2 - 54x + 18x - 54)/(x + 3)^2(x - 3)

Combining like terms again: (-x^3 + 26x^2 - 36x - 54)/(x + 3)^2(x - 3)

Therefore, the simplified expression is: (-x^3 + 26x^2 - 36x - 54)/(x + 3)^2(x - 3)