Express in simplest form.

1. -4/x^3 + 9/x + 2/x^2
2. 7/10x^2 + 1/2x^3 + 11/5x
3. a + 4/3a + 2a - 1/5a^2

I wonder if a^-3 is simpler than 1/a^3

I have never in my life heard it was, perhaps your teacher went to some school that taught such nonsense.

in 3) there are two terms that can be combined to give 3a

Otherwise, I don't see anything simpler, perhaps I have missed something.

To express the given expressions in their simplest form, we will simplify the fractions and combine like terms.

1. -4/x^3 + 9/x + 2/x^2:
To add fractions, we need to find a common denominator. In this case, the smallest common denominator is x^3. We rewrite each fraction with the common denominator:
-4/x^3 + 9/x + 2/x^2 = (-4 * x^2) / (x^3 * x^2) + (9 * x^3) / (x * x^3) + (2 * x) / (x^2 * x).
Simplifying further:
= (-4x^2) / (x^5) + (9x^3) / (x^4) + (2x) / (x^3).
Now we can combine the fractions by adding the numerators and keeping the denominator common:
= (-4x^2 + 9x^3 + 2x) / (x^5 + x^4 + x^3).
This is the simplest form of the expression.

2. 7/10x^2 + 1/2x^3 + 11/5x:
In this expression, we don't have any common factors in the denominators. To add the fractions, we first need to find a common denominator. The smallest common denominator is 10x^2 * 2x^3 * 5x, which simplifies to 100x^6.
Rewriting each fraction with the common denominator:
= (7 * 2x^3 * 5x) / (10x^2 * 2x^3 * 5x) + (1 * 10x^2 * 5x) / (2x^3 * 10x^2 * 5x) + (11 * 2x^2 * 2x^3) / (5x * 2x^2 * 2x^3).
Simplifying further:
= (70x^4) / (100x^6) + (50x^3) / (100x^6) + (44x^5) / (20x^6).
Now we can add the fractions by combining the numerators while keeping the denominator common:
= (70x^4 + 50x^3 + 44x^5) / (100x^6).
This is the simplest form of the expression.

3. a + 4/3a + 2a - 1/5a^2:
First, let's combine like terms. We can add the terms with just 'a' together:
= a + 2a + 4/3a - 1/5a^2.
Now, we can deal with the fractions separately. To add the fractions, we need to find a common denominator. The smallest common denominator in this case is 15a^2:
= a + 2a + (4 * 5) / (3 * 5a^2) - (1 * 3) / (5 * 3a^2).
Simplifying further:
= a + 2a + 20/15a^2 - 3/15a^2.
Now we combine like terms by adding the coefficients of 'a' and the fractions:
= (1 + 2)a + (20 - 3) / 15a^2.
Simplifying:
= 3a + 17/15a^2.
This is the simplest form of the expression.