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Answer :
To fully simplify the given expression and write it as a single fraction, we need to work through each part of the expression step by step:
### Step 1: Simplify each fraction
1. Fraction 1: [tex]\(\frac{x^4 - 4x^2}{5x^5 + 45x^4 + 70x^3}\)[/tex]
- Numerator: [tex]\(x^4 - 4x^2\)[/tex] can be factored as [tex]\(x^2(x^2 - 4)\)[/tex]. Notice [tex]\(x^2 - 4\)[/tex] can be further factored as [tex]\((x - 2)(x + 2)\)[/tex].
- So, it becomes [tex]\(x^2(x - 2)(x + 2)\)[/tex].
- Denominator: [tex]\(5x^5 + 45x^4 + 70x^3\)[/tex] can be factored by first taking out a common factor [tex]\(x^3\)[/tex], giving [tex]\(x^3(5x^2 + 45x + 70)\)[/tex].
- Further factorization of the quadratic term requires specific methods (e.g., trial and error, quadratic formula). Here, it can be factored as [tex]\((5x + 7)(x + 10)\)[/tex].
2. Fraction 2: [tex]\(\frac{5x + 35}{x + 10}\)[/tex]
- Numerator: [tex]\(5x + 35\)[/tex] can be factored as [tex]\(5(x + 7)\)[/tex].
3. Fraction 3: [tex]\(\frac{x - 2}{x^2 + 10}\)[/tex]
- This fraction is already simplified since [tex]\(x^2 + 10\)[/tex] cannot be factored further using real numbers.
4. Fraction 4: [tex]\(\frac{x - 2}{x + 10}\)[/tex]
- This fraction is already in its simplest form.
### Step 2: Multiply all fractions
Combine all the fractions into one by multiplying numerators and denominators:
- Numerator: [tex]\(x^2(x - 2)(x + 2) \cdot 5(x + 7) \cdot (x - 2)\)[/tex]
- Denominator: [tex]\(x^3(5x + 7)(x + 10) \cdot (x + 10) \cdot (x^2 + 10)\)[/tex]
### Step 3: Simplify the resulting expression
1. Combine numerator and simplify:
- [tex]\(x^2(x - 2)^2(x + 2) \cdot 5(x + 7)\)[/tex]
2. Combine denominator and simplify:
- [tex]\(x^3(5x + 7)(x + 10)^2(x^2 + 10)\)[/tex]
Simplify by canceling any common factors:
- Factor [tex]\((x + 7)\)[/tex] and others that might cancel out.
### Result
After carefully following all the above steps, this simplifies to:
[tex]\[
\frac{(x - 2)^2(x + 7)(x^2 - 4)}{x(x + 10)^2(x^2 + 10)(x^2 + 9x + 14)}
\][/tex]
This is the simplified form of the given expression, written as a single fraction.
### Step 1: Simplify each fraction
1. Fraction 1: [tex]\(\frac{x^4 - 4x^2}{5x^5 + 45x^4 + 70x^3}\)[/tex]
- Numerator: [tex]\(x^4 - 4x^2\)[/tex] can be factored as [tex]\(x^2(x^2 - 4)\)[/tex]. Notice [tex]\(x^2 - 4\)[/tex] can be further factored as [tex]\((x - 2)(x + 2)\)[/tex].
- So, it becomes [tex]\(x^2(x - 2)(x + 2)\)[/tex].
- Denominator: [tex]\(5x^5 + 45x^4 + 70x^3\)[/tex] can be factored by first taking out a common factor [tex]\(x^3\)[/tex], giving [tex]\(x^3(5x^2 + 45x + 70)\)[/tex].
- Further factorization of the quadratic term requires specific methods (e.g., trial and error, quadratic formula). Here, it can be factored as [tex]\((5x + 7)(x + 10)\)[/tex].
2. Fraction 2: [tex]\(\frac{5x + 35}{x + 10}\)[/tex]
- Numerator: [tex]\(5x + 35\)[/tex] can be factored as [tex]\(5(x + 7)\)[/tex].
3. Fraction 3: [tex]\(\frac{x - 2}{x^2 + 10}\)[/tex]
- This fraction is already simplified since [tex]\(x^2 + 10\)[/tex] cannot be factored further using real numbers.
4. Fraction 4: [tex]\(\frac{x - 2}{x + 10}\)[/tex]
- This fraction is already in its simplest form.
### Step 2: Multiply all fractions
Combine all the fractions into one by multiplying numerators and denominators:
- Numerator: [tex]\(x^2(x - 2)(x + 2) \cdot 5(x + 7) \cdot (x - 2)\)[/tex]
- Denominator: [tex]\(x^3(5x + 7)(x + 10) \cdot (x + 10) \cdot (x^2 + 10)\)[/tex]
### Step 3: Simplify the resulting expression
1. Combine numerator and simplify:
- [tex]\(x^2(x - 2)^2(x + 2) \cdot 5(x + 7)\)[/tex]
2. Combine denominator and simplify:
- [tex]\(x^3(5x + 7)(x + 10)^2(x^2 + 10)\)[/tex]
Simplify by canceling any common factors:
- Factor [tex]\((x + 7)\)[/tex] and others that might cancel out.
### Result
After carefully following all the above steps, this simplifies to:
[tex]\[
\frac{(x - 2)^2(x + 7)(x^2 - 4)}{x(x + 10)^2(x^2 + 10)(x^2 + 9x + 14)}
\][/tex]
This is the simplified form of the given expression, written as a single fraction.
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