Simplify:$\frac{25 W^5}{45 W^4}$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved in simplifying fractions with variables. In this article, we will focus on simplifying the given expression $\frac{25 w^5}{45 w^4}$ using the rules of exponents and fraction simplification.

Understanding the Expression

The given expression is a fraction with variables in the numerator and denominator. The numerator is $25 w^5$, and the denominator is $45 w^4$. To simplify this expression, we need to apply the rules of exponents and fraction simplification.

Simplifying the Numerator and Denominator

To simplify the expression, we can start by simplifying the numerator and denominator separately. The numerator is $25 w^5$, and the denominator is $45 w^4$. We can simplify the numerator by factoring out the greatest common factor (GCF) of the coefficients and the variables.

Factoring Out the GCF

The GCF of the coefficients 25 and 45 is 5. The GCF of the variables $w^5$ and $w^4$ is $w^4$. Therefore, we can factor out the GCF of the coefficients and the variables as follows:

$$\frac{25 w^5}{45 w^4} = \frac{5 \cdot 5 \cdot w^4 \cdot w}{5 \cdot 9 \cdot w^4}$$

Canceling Out Common Factors

Now that we have factored out the GCF, we can cancel out the common factors in the numerator and denominator. The common factors are $5$, $w^4$, and $w$. Canceling out these common factors, we get:

$$\frac{5 \cdot 5 \cdot w^4 \cdot w}{5 \cdot 9 \cdot w^4} = \frac{5 \cdot w}{9}$$

Simplifying the Expression

Now that we have canceled out the common factors, we can simplify the expression further. We can simplify the expression by dividing the numerator and denominator by their greatest common factor (GCF).

Dividing by the GCF

The GCF of the coefficients 5 and 9 is 1. Therefore, we can divide the numerator and denominator by 1, which leaves us with:

$$\frac{5 \cdot w}{9}$$

Conclusion

In conclusion, we have simplified the given expression $\frac{25 w^5}{45 w^4}$ using the rules of exponents and fraction simplification. We factored out the GCF of the coefficients and the variables, canceled out the common factors, and simplified the expression further by dividing the numerator and denominator by their GCF. The simplified expression is $\frac{5 \cdot w}{9}$.

Final Answer

The final answer is $\boxed{\frac{5w}{9}}$.

Applications of Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous applications in various fields such as physics, engineering, and computer science. Some of the applications of simplifying algebraic expressions include:

• Solving Equations: Simplifying algebraic expressions is essential in solving equations. By simplifying the expressions, we can isolate the variable and solve for its value.
• Graphing Functions: Simplifying algebraic expressions is also essential in graphing functions. By simplifying the expressions, we can identify the x-intercepts, y-intercepts, and other key features of the graph.
• Optimization: Simplifying algebraic expressions is also essential in optimization problems. By simplifying the expressions, we can identify the maximum or minimum value of a function.
• Computer Science: Simplifying algebraic expressions is also essential in computer science. By simplifying the expressions, we can optimize algorithms and improve the performance of computer programs.

Tips for Simplifying Algebraic Expressions

Simplifying algebraic expressions can be challenging, but there are several tips that can help. Some of the tips for simplifying algebraic expressions include:

• Identify the GCF: The greatest common factor (GCF) is the largest factor that divides both the numerator and denominator. Identifying the GCF is essential in simplifying algebraic expressions.
• Cancel Out Common Factors: Canceling out common factors is essential in simplifying algebraic expressions. By canceling out common factors, we can simplify the expression and make it easier to work with.
• Use the Rules of Exponents: The rules of exponents are essential in simplifying algebraic expressions. By applying the rules of exponents, we can simplify the expression and make it easier to work with.
• Practice, Practice, Practice: Simplifying algebraic expressions requires practice. By practicing regularly, we can develop the skills and techniques needed to simplify complex expressions.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous applications in various fields such as physics, engineering, and computer science. By understanding the rules of exponents and fraction simplification, we can simplify complex expressions and make them easier to work with. By practicing regularly, we can develop the skills and techniques needed to simplify complex expressions and become proficient in simplifying algebraic expressions.

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Introduction

In our previous article, we simplified the given expression $\frac{25 w^5}{45 w^4}$ using the rules of exponents and fraction simplification. In this article, we will answer some of the frequently asked questions related to simplifying algebraic expressions.

Q&A

Q1: What is the greatest common factor (GCF) of the coefficients 25 and 45?

A1: The greatest common factor (GCF) of the coefficients 25 and 45 is 5.

Q2: What is the greatest common factor (GCF) of the variables $w^5$ and $w^4$?

A2: The greatest common factor (GCF) of the variables $w^5$ and $w^4$ is $w^4$.

Q3: How do I simplify the expression $\frac{25 w^5}{45 w^4}$?

A3: To simplify the expression $\frac{25 w^5}{45 w^4}$, you can factor out the greatest common factor (GCF) of the coefficients and the variables, cancel out the common factors, and simplify the expression further by dividing the numerator and denominator by their greatest common factor (GCF).

Q4: What is the simplified expression of $\frac{25 w^5}{45 w^4}$?

A4: The simplified expression of $\frac{25 w^5}{45 w^4}$ is $\frac{5 \cdot w}{9}$.

Q5: How do I identify the greatest common factor (GCF) of the coefficients and the variables?

A5: To identify the greatest common factor (GCF) of the coefficients and the variables, you can list the factors of each coefficient and variable, and then find the greatest common factor among them.

Q6: What are some of the applications of simplifying algebraic expressions?

A6: Some of the applications of simplifying algebraic expressions include solving equations, graphing functions, optimization, and computer science.

Q7: How do I simplify complex expressions?

A7: To simplify complex expressions, you can use the rules of exponents and fraction simplification, factor out the greatest common factor (GCF) of the coefficients and the variables, cancel out the common factors, and simplify the expression further by dividing the numerator and denominator by their greatest common factor (GCF).

Q8: What are some of the tips for simplifying algebraic expressions?

A8: Some of the tips for simplifying algebraic expressions include identifying the greatest common factor (GCF), canceling out common factors, using the rules of exponents, and practicing regularly.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous applications in various fields such as physics, engineering, and computer science. By understanding the rules of exponents and fraction simplification, we can simplify complex expressions and make them easier to work with. By practicing regularly, we can develop the skills and techniques needed to simplify complex expressions and become proficient in simplifying algebraic expressions.

Final Answer

The final answer is $\boxed{\frac{5w}{9}}$.

Additional Resources

For more information on simplifying algebraic expressions, you can refer to the following resources:

• Simplifying Algebraic Expressions
• Rules of Exponents
• Fraction Simplification

Related Articles

For more information on algebraic expressions, you can refer to the following articles:

• Simplifying Algebraic Expressions
• Rules of Exponents
• Fraction Simplification

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