Match each subtraction expression with its answer.

1. [tex]\(\frac{2}{a^2 b^3}-\frac{1}{a b}\)[/tex]
[tex]\(\square\)[/tex]

2. [tex]\(\frac{1}{a b^2}-\frac{1}{a^3 b}\)[/tex]
[tex]\(\square\)[/tex]

3. [tex]\(\frac{2}{a^3 b^3}-\frac{1}{a^2 b^3}\)[/tex]
[tex]\(\square\)[/tex]

4. [tex]\(\frac{1}{a^3 b^2}-\frac{1}{a^2 b^3}\)[/tex]
[tex]\(\square\)[/tex]

Answers:
A. [tex]\(\frac{a^2 - b}{a^3 b^2}\)[/tex]
B. [tex]\(\frac{b - a}{a^9 b^9}\)[/tex]
C. [tex]\(\frac{2 - ab^2}{a^2 b^3}\)[/tex]
D. [tex]\(\frac{2 - a}{a^3 b^3}\)[/tex]



Answer :

Certainly! Let's work through each of the given expressions step by step and provide the simplified results.

### 1. [tex]\(\frac{2}{a^2 b^3} - \frac{1}{a b}\)[/tex]

First, let's simplify the fraction:

[tex]\[ \frac{2}{a^2 b^3} - \frac{1}{a b} \][/tex]

To combine these fractions, we need a common denominator, which is [tex]\(a^2 b^3\)[/tex]:

[tex]\[ \frac{2}{a^2 b^3} - \frac{1 \cdot a}{a \cdot a b^3} = \frac{2}{a^2 b^3} - \frac{a b^2}{a^2 b^3} \][/tex]

Now, subtract the numerators:

[tex]\[ = \frac{2 - a b^2}{a^2 b^3} \][/tex]

So, the simplified expression is:

[tex]\[ \frac{2}{a^2 b^3} - \frac{1}{a b} = \frac{2 - a b^2}{a^2 b^3} \][/tex]

### 2. [tex]\(\frac{1}{a b^2} - \frac{1}{a^3 b}\)[/tex]

Next, we'll handle the second expression:

[tex]\[ \frac{1}{a b^2} - \frac{1}{a^3 b} \][/tex]

Find a common denominator, which is [tex]\(a^3 b^2\)[/tex]:

[tex]\[ \frac{1 \cdot a^2}{a^2 \cdot a b^2} - \frac{1 \cdot b}{a^3 \cdot b b} = \frac{a^2}{a^3 b^2} - \frac{b}{a^3 b^2} \][/tex]

Now, subtract the numerators:

[tex]\[ = \frac{a^2 - b}{a^3 b^2} \][/tex]

So, the simplified expression is:

[tex]\[ \frac{1}{a b^2} - \frac{1}{a^3 b} = \frac{a^2 - b}{a^3 b^2} \][/tex]

### 3. [tex]\(\frac{2}{a^3 b^3} - \frac{1}{a^2 b^3}\)[/tex]

Simplify the third expression:

[tex]\[ \frac{2}{a^3 b^3} - \frac{1}{a^2 b^3} \][/tex]

Find a common denominator, which is [tex]\(a^3 b^3\)[/tex]:

[tex]\[ \frac{2}{a^3 b^3} - \frac{1 \cdot a}{a \cdot a^2 b^3} = \frac{2}{a^3 b^3} - \frac{a}{a^3 b^3} \][/tex]

Now, subtract the numerators:

[tex]\[ = \frac{2 - a}{a^3 b^3} \][/tex]

So, the simplified expression is:

[tex]\[ \frac{2}{a^3 b^3} - \frac{1}{a^2 b^3} = \frac{2 - a}{a^3 b^3} \][/tex]

### 4. [tex]\(\frac{1}{a^3 b^2} - \frac{1}{a^2 b^3}\)[/tex]

Simplify the fourth expression:

[tex]\[ \frac{1}{a^3 b^2} - \frac{1}{a^2 b^3} \][/tex]

Find a common denominator, which is [tex]\(a^3 b^3\)[/tex]:

[tex]\[ \frac{1 \cdot b}{a^3 \cdot b b^2} - \frac{1 \cdot a}{a \cdot a^2 b^3} = \frac{b}{a^3 b^3} - \frac{a}{a^3 b^3} \][/tex]

Now, subtract the numerators:

[tex]\[ = \frac{b - a}{a^3 b^3} \][/tex]

So, the simplified expression is:

[tex]\[ \frac{1}{a^3 b^2} - \frac{1}{a^2 b^3} = \frac{b - a}{a^3 b^3} \][/tex]

### 5. [tex]\(\frac{a^2 - b}{a^3 b^2}\)[/tex]

This expression is already in its simplified form:

[tex]\[ \frac{a^2 - b}{a^3 b^2} \][/tex]

### 6. [tex]\(\frac{b - a}{a^9 b^9}\)[/tex]

This expression is also already simplified in its simplest form:

[tex]\[ \frac{b - a}{a^9 b^9} \][/tex]

### 7. [tex]\(\frac{2 - a b^2}{a^2 b^3}\)[/tex]

This expression is a straightforward fraction and is already in its simplest form:

[tex]\[ \frac{2 - a b^2}{a^2 b^3} \][/tex]

### 8. [tex]\(\frac{2 - a}{a^3 b^3}\)[/tex]

Finally, this expression is also a simplified fraction:

[tex]\[ \frac{2 - a}{a^3 b^3} \][/tex]

Thus, the final simplified forms for these expressions are:

1. [tex]\(\frac{2}{a^2 b^3} - \frac{1}{a b} = \frac{2 - a b^2}{a^2 b^3}\)[/tex]
2. [tex]\(\frac{1}{a b^2} - \frac{1}{a^3 b} = \frac{a^2 - b}{a^3 b^2}\)[/tex]
3. [tex]\(\frac{2}{a^3 b^3} - \frac{1}{a^2 b^3} = \frac{2 - a}{a^3 b^3}\)[/tex]
4. [tex]\(\frac{1}{a^3 b^2} - \frac{1}{a^2 b^3} = \frac{b - a}{a^3 b^3}\)[/tex]
5. [tex]\(\frac{a^2 - b}{a^3 b^2}\)[/tex]
6. [tex]\(\frac{b - a}{a^9 b^9}\)[/tex]
7. [tex]\(\frac{2 - a b^2}{a^2 b^3}\)[/tex]
8. [tex]\(\frac{2 - a}{a^3 b^3}\)[/tex]

I hope this step-by-step solution is clear and helpful!