Answer :

Sure! Let's simplify the given expression step by step and write the final answer with positive exponents.

The given expression is:
[tex]\[
\frac{-18x^7 - 21x^5 - 9x^3}{-3x^5}
\][/tex]

### Step 1: Break down the expression
We can split the expression into three separate fractions:
[tex]\[
\frac{-18x^7}{-3x^5} + \frac{-21x^5}{-3x^5} + \frac{-9x^3}{-3x^5}
\][/tex]

### Step 2: Simplify each fraction separately

#### First fraction: [tex]\(\frac{-18x^7}{-3x^5}\)[/tex]
1. Divide the coefficients: [tex]\(\frac{-18}{-3} = 6\)[/tex]
2. Subtract the exponents of [tex]\(x\)[/tex]: [tex]\(x^{7-5} = x^2\)[/tex]

So, the first fraction simplifies to:
[tex]\[
6x^2
\][/tex]

#### Second fraction: [tex]\(\frac{-21x^5}{-3x^5}\)[/tex]
1. Divide the coefficients: [tex]\(\frac{-21}{-3} = 7\)[/tex]
2. Subtract the exponents of [tex]\(x\)[/tex]: [tex]\(x^{5-5} = x^0 = 1\)[/tex]

So, the second fraction simplifies to:
[tex]\[
7
\][/tex]

#### Third fraction: [tex]\(\frac{-9x^3}{-3x^5}\)[/tex]
1. Divide the coefficients: [tex]\(\frac{-9}{-3} = 3\)[/tex]
2. Subtract the exponents of [tex]\(x\)[/tex]: [tex]\(x^{3-5} = x^{-2}\)[/tex]

So, the third fraction simplifies to:
[tex]\[
3x^{-2}
\][/tex]

### Step 3: Combine the simplified fractions
Now, let's put all the simplified fractions together:
[tex]\[
6x^2 + 7 + 3x^{-2}
\][/tex]

### Step 4: Write with positive exponents
The term [tex]\(3x^{-2}\)[/tex] can be written with a positive exponent as:
[tex]\[
3 \cdot \frac{1}{x^2} = \frac{3}{x^2}
\][/tex]

So, the final simplified expression is:
[tex]\[
6x^2 + 7 + \frac{3}{x^2}
\][/tex]

Therefore, the simplified expression with positive exponents is:
[tex]\[
6x^2 + 7 + \frac{3}{x^2}
\][/tex]

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Rewritten by : Barada