Simplify The Expression: 6 S 2 + 9 S S 2 \frac{6s^2 + 9s}{s^2} S 2 6 S 2 + 9 S ​

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves rewriting an expression in a simpler form, often by combining like terms or canceling out common factors. In this article, we will focus on simplifying the expression $\frac{6s^2 + 9s}{s^2}$ using various techniques.

Understanding the Expression

The given expression is a rational expression, which is a fraction that contains variables and constants in the numerator and denominator. In this case, the numerator is $6s^2 + 9s$ and the denominator is $s^2$. To simplify this expression, we need to analyze the numerator and denominator separately.

Factoring the Numerator

The numerator $6s^2 + 9s$ can be factored by taking out the greatest common factor (GCF), which is $3s$. Factoring out $3s$ from both terms, we get:

$$6s^2 + 9s = 3s(2s + 3)$$

Simplifying the Expression

Now that we have factored the numerator, we can rewrite the original expression as:

$$\frac{6s^2 + 9s}{s^2} = \frac{3s(2s + 3)}{s^2}$$

Canceling Out Common Factors

We can simplify the expression further by canceling out the common factor of $s$ in the numerator and denominator. This leaves us with:

$$\frac{3s(2s + 3)}{s^2} = \frac{3(2s + 3)}{s}$$

Final Simplification

The expression $\frac{3(2s + 3)}{s}$ can be simplified by distributing the $3$ to the terms inside the parentheses:

$$\frac{3(2s + 3)}{s} = \frac{6s + 9}{s}$$

Combining Like Terms

The final expression $\frac{6s + 9}{s}$ can be simplified by combining the like terms in the numerator. However, in this case, there are no like terms to combine, so the expression remains the same.

Conclusion

In conclusion, we have simplified the expression $\frac{6s^2 + 9s}{s^2}$ using various techniques, including factoring, canceling out common factors, and combining like terms. The final simplified expression is $\frac{6s + 9}{s}$.

Tips and Tricks

• When simplifying rational expressions, it's essential to factor the numerator and denominator separately.
• Canceling out common factors can help simplify the expression, but be careful not to cancel out any important terms.
• Combining like terms can help simplify the expression, but only if there are like terms to combine.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

• Science and Engineering: Simplifying expressions is crucial in scientific and engineering applications, where complex equations need to be solved to model real-world phenomena.
• Finance: Simplifying expressions is essential in finance, where complex financial models need to be solved to make informed investment decisions.
• Computer Science: Simplifying expressions is critical in computer science, where complex algorithms need to be optimized to improve performance.

Common Mistakes to Avoid

When simplifying rational expressions, it's essential to avoid common mistakes, including:

• Canceling out important terms: Be careful not to cancel out any important terms, as this can lead to incorrect solutions.
• Not factoring the numerator and denominator: Failing to factor the numerator and denominator can lead to incorrect simplifications.
• Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By understanding the techniques and strategies outlined in this article, you can simplify complex expressions and solve real-world problems with confidence. Remember to factor the numerator and denominator, cancel out common factors, and combine like terms to simplify expressions. With practice and patience, you'll become proficient in simplifying algebraic expressions and tackle complex problems with ease.

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Introduction

In our previous article, we simplified the expression $\frac{6s^2 + 9s}{s^2}$ using various techniques, including factoring, canceling out common factors, and combining like terms. In this article, we will answer some frequently asked questions (FAQs) related to simplifying rational expressions.

Q&A

Q: What is the first step in simplifying a rational expression?

A: The first step in simplifying a rational expression is to factor the numerator and denominator separately.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, look for the greatest common factor (GCF) and factor it out. You can also use the distributive property to factor the numerator and denominator.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves breaking down an expression into simpler components, while canceling out common factors involves eliminating common factors between the numerator and denominator.

Q: Can I cancel out any term in the numerator and denominator?

A: No, you can only cancel out common factors between the numerator and denominator. Be careful not to cancel out any important terms.

Q: How do I combine like terms in a rational expression?

A: To combine like terms in a rational expression, look for terms with the same variable and exponent. You can then add or subtract these terms.

Q: What is the final simplified expression for $\frac{6s^2 + 9s}{s^2}$?

A: The final simplified expression for $\frac{6s^2 + 9s}{s^2}$ is $\frac{6s + 9}{s}$.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, be careful not to cancel out any important terms.

Q: How do I know if a rational expression is already simplified?

A: A rational expression is already simplified if there are no common factors between the numerator and denominator, and there are no like terms to combine.

Q: Can I use a calculator to simplify a rational expression?

A: Yes, you can use a calculator to simplify a rational expression. However, it's essential to understand the underlying math and be able to simplify expressions manually.

Tips and Tricks

• Always factor the numerator and denominator separately before simplifying a rational expression.
• Be careful not to cancel out any important terms when simplifying a rational expression.
• Combine like terms in a rational expression to simplify it further.
• Use a calculator to check your work and ensure that the expression is simplified correctly.

Real-World Applications

Simplifying rational expressions has numerous real-world applications, including:

• Science and Engineering: Simplifying expressions is crucial in scientific and engineering applications, where complex equations need to be solved to model real-world phenomena.
• Finance: Simplifying expressions is essential in finance, where complex financial models need to be solved to make informed investment decisions.
• Computer Science: Simplifying expressions is critical in computer science, where complex algorithms need to be optimized to improve performance.

Common Mistakes to Avoid

When simplifying rational expressions, it's essential to avoid common mistakes, including:

• Canceling out important terms: Be careful not to cancel out any important terms, as this can lead to incorrect solutions.
• Not factoring the numerator and denominator: Failing to factor the numerator and denominator can lead to incorrect simplifications.
• Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.

Final Thoughts

Simplifying rational expressions is a crucial skill in mathematics, particularly in algebra and calculus. By understanding the techniques and strategies outlined in this article, you can simplify complex expressions and solve real-world problems with confidence. Remember to factor the numerator and denominator, cancel out common factors, and combine like terms to simplify expressions. With practice and patience, you'll become proficient in simplifying rational expressions and tackle complex problems with ease.