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Answer :
- Divide each term in the numerator by the denominator: $\frac{5 x^9}{5 x^3} + \frac{15 x^6}{5 x^3} - \frac{20 x^3}{5 x^3} + \frac{2 x}{5 x^3}$.
- Simplify each term using the quotient rule for exponents: $x^{9-3} + 3x^{6-3} - 4x^{3-3} + \frac{2}{5}x^{1-3}$.
- Simplify the exponents: $x^6 + 3x^3 - 4 + \frac{2}{5}x^{-2}$.
- Express the final simplified form: $\boxed{x^6 + 3x^3 - 4 + \frac{2}{5x^2}}$.
### Explanation
1. Understanding the Problem
We are asked to simplify the expression $\frac{5 x^9+15 x^6-20 x^3+2 x}{5 x^3}$. This involves dividing each term in the numerator by the denominator.
2. Dividing Each Term
We divide each term in the numerator by $5x^3$:$$\frac{5 x^9}{5 x^3} + \frac{15 x^6}{5 x^3} - \frac{20 x^3}{5 x^3} + \frac{2 x}{5 x^3}$$
3. Simplifying Each Term
Now, we simplify each term using the quotient rule for exponents, which states that $\frac{x^a}{x^b} = x^{a-b}$:$$\frac{5 x^9}{5 x^3} = x^{9-3} = x^6$$$$\frac{15 x^6}{5 x^3} = 3x^{6-3} = 3x^3$$$$\frac{20 x^3}{5 x^3} = 4x^{3-3} = 4$$$$\frac{2 x}{5 x^3} = \frac{2}{5}x^{1-3} = \frac{2}{5}x^{-2} = \frac{2}{5x^2}$$
4. Combining Terms
Combining the simplified terms, we get:$$x^6 + 3x^3 - 4 + \frac{2}{5x^2}$$
5. Final Answer
Therefore, the simplified expression is $\boxed{x^6 + 3x^3 - 4 + \frac{2}{5x^2}}$.
### Examples
Simplifying rational expressions is useful in various fields, such as physics and engineering, where complex equations can be made easier to work with by simplifying them. For example, when calculating the trajectory of a projectile, simplifying the equations can make the calculations more manageable and provide a clearer understanding of the factors affecting the projectile's motion. It also helps in optimizing designs and predicting outcomes in complex systems.
- Simplify each term using the quotient rule for exponents: $x^{9-3} + 3x^{6-3} - 4x^{3-3} + \frac{2}{5}x^{1-3}$.
- Simplify the exponents: $x^6 + 3x^3 - 4 + \frac{2}{5}x^{-2}$.
- Express the final simplified form: $\boxed{x^6 + 3x^3 - 4 + \frac{2}{5x^2}}$.
### Explanation
1. Understanding the Problem
We are asked to simplify the expression $\frac{5 x^9+15 x^6-20 x^3+2 x}{5 x^3}$. This involves dividing each term in the numerator by the denominator.
2. Dividing Each Term
We divide each term in the numerator by $5x^3$:$$\frac{5 x^9}{5 x^3} + \frac{15 x^6}{5 x^3} - \frac{20 x^3}{5 x^3} + \frac{2 x}{5 x^3}$$
3. Simplifying Each Term
Now, we simplify each term using the quotient rule for exponents, which states that $\frac{x^a}{x^b} = x^{a-b}$:$$\frac{5 x^9}{5 x^3} = x^{9-3} = x^6$$$$\frac{15 x^6}{5 x^3} = 3x^{6-3} = 3x^3$$$$\frac{20 x^3}{5 x^3} = 4x^{3-3} = 4$$$$\frac{2 x}{5 x^3} = \frac{2}{5}x^{1-3} = \frac{2}{5}x^{-2} = \frac{2}{5x^2}$$
4. Combining Terms
Combining the simplified terms, we get:$$x^6 + 3x^3 - 4 + \frac{2}{5x^2}$$
5. Final Answer
Therefore, the simplified expression is $\boxed{x^6 + 3x^3 - 4 + \frac{2}{5x^2}}$.
### Examples
Simplifying rational expressions is useful in various fields, such as physics and engineering, where complex equations can be made easier to work with by simplifying them. For example, when calculating the trajectory of a projectile, simplifying the equations can make the calculations more manageable and provide a clearer understanding of the factors affecting the projectile's motion. It also helps in optimizing designs and predicting outcomes in complex systems.
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