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Sagot :
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!
### 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!
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