Simplify The Expression: ${ \frac{4a^2 - 40a}{9a^3 - 72a^2 - 180a} }$

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Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves reducing complex expressions to their simplest form, making it easier to solve equations and manipulate variables. In this article, we will focus on simplifying the given expression: $\frac{4a^2 - 40a}{9a^3 - 72a^2 - 180a}$. We will use various techniques, such as factoring, canceling, and combining like terms, to simplify the expression.

Factoring the Numerator and Denominator

To simplify the expression, we need to factor the numerator and denominator. The numerator can be factored as follows:

$$4a^2 - 40a = 4a(a - 10)$$

The denominator can be factored as follows:

$$9a^3 - 72a^2 - 180a = 9a(a^2 - 8a - 20)$$

Factoring the Quadratic Expression in the Denominator

The quadratic expression in the denominator can be factored as follows:

$$a^2 - 8a - 20 = (a - 10)(a + 2)$$

Canceling Common Factors

Now that we have factored the numerator and denominator, we can cancel common factors. The expression can be rewritten as:

$$\frac{4a(a - 10)}{9a(a - 10)(a + 2)}$$

We can cancel the common factor $(a - 10)$ from the numerator and denominator:

$$\frac{4a}{9a(a + 2)}$$

Canceling the Common Factor $a$

We can also cancel the common factor $a$ from the numerator and denominator:

$$\frac{4}{9(a + 2)}$$

Conclusion

In conclusion, we have simplified the given expression: $\frac{4a^2 - 40a}{9a^3 - 72a^2 - 180a}$. We used various techniques, such as factoring, canceling, and combining like terms, to simplify the expression. The final simplified expression is $\frac{4}{9(a + 2)}$.

Tips for Simplifying Algebraic Expressions

Here are some tips for simplifying algebraic expressions:

• Factor the numerator and denominator: Factoring the numerator and denominator can help you identify common factors that can be canceled.
• Cancel common factors: Canceling common factors can help you simplify the expression.
• Combine like terms: Combining like terms can help you simplify the expression.
• Use the distributive property: The distributive property can help you expand and simplify expressions.
• Use the commutative property: The commutative property can help you rearrange terms and simplify expressions.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying algebraic expressions:

• Not factoring the numerator and denominator: Failing to factor the numerator and denominator can make it difficult to identify common factors.
• Not canceling common factors: Failing to cancel common factors can result in a more complex expression.
• Not combining like terms: Failing to combine like terms can result in a more complex expression.
• Not using the distributive property: Failing to use the distributive property can make it difficult to expand and simplify expressions.
• Not using the commutative property: Failing to use the commutative property can make it difficult to rearrange terms and simplify expressions.

Real-World Applications of Simplifying Algebraic Expressions

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

• Physics and engineering: Simplifying algebraic expressions is essential in physics and engineering, where complex equations need to be solved to model real-world phenomena.
• Computer science: Simplifying algebraic expressions is essential in computer science, where complex algorithms need to be optimized to improve performance.
• Economics: Simplifying algebraic expressions is essential in economics, where complex models need to be solved to understand economic phenomena.
• Biology: Simplifying algebraic expressions is essential in biology, where complex models need to be solved to understand biological phenomena.

Final Thoughts

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By using various techniques, such as factoring, canceling, and combining like terms, we can simplify complex expressions and make them easier to solve. Remember to factor the numerator and denominator, cancel common factors, combine like terms, use the distributive property, and use the commutative property to simplify algebraic expressions.

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Introduction

In our previous article, we simplified the expression: $\frac{4a^2 - 40a}{9a^3 - 72a^2 - 180a}$. We used various techniques, such as factoring, canceling, and combining like terms, to simplify the expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factor the numerator and denominator. This can help you identify common factors that can be canceled.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you can use various techniques, such as factoring out common factors, using the distributive property, and using the commutative property.

Q: What is the difference between factoring and canceling?

A: Factoring involves breaking down an expression into its component parts, while canceling involves eliminating common factors between the numerator and denominator.

Q: How do I know when to cancel common factors?

A: You should cancel common factors when you have factored the numerator and denominator and have identified common factors that can be eliminated.

Q: What is the importance of combining like terms?

A: Combining like terms is essential in simplifying algebraic expressions. It helps to eliminate unnecessary terms and make the expression easier to solve.

Q: How do I combine like terms?

A: To combine like terms, you can add or subtract the coefficients of the terms with the same variable.

Q: What is the distributive property, and how do I use it?

A: The distributive property is a mathematical property that allows you to multiply a single term by multiple terms. You can use it to expand and simplify expressions.

Q: What is the commutative property, and how do I use it?

A: The commutative property is a mathematical property that allows you to rearrange terms in an expression. You can use it to simplify expressions and make them easier to solve.

Q: How do I know when an expression is simplified?

A: An expression is simplified when you have eliminated all unnecessary terms and have factored the numerator and denominator.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include not factoring the numerator and denominator, not canceling common factors, not combining like terms, and not using the distributive property and commutative property.

Real-World Applications of Simplifying Algebraic Expressions

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

• Physics and engineering: Simplifying algebraic expressions is essential in physics and engineering, where complex equations need to be solved to model real-world phenomena.
• Computer science: Simplifying algebraic expressions is essential in computer science, where complex algorithms need to be optimized to improve performance.
• Economics: Simplifying algebraic expressions is essential in economics, where complex models need to be solved to understand economic phenomena.
• Biology: Simplifying algebraic expressions is essential in biology, where complex models need to be solved to understand biological phenomena.

Final Thoughts

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By using various techniques, such as factoring, canceling, and combining like terms, we can simplify complex expressions and make them easier to solve. Remember to factor the numerator and denominator, cancel common factors, combine like terms, use the distributive property, and use the commutative property to simplify algebraic expressions.

Additional Resources

If you need additional help with simplifying algebraic expressions, here are some additional resources:

• Online tutorials: There are many online tutorials and videos that can help you learn how to simplify algebraic expressions.
• Math textbooks: Math textbooks can provide you with a comprehensive guide to simplifying algebraic expressions.
• Math software: Math software, such as Mathematica and Maple, can help you simplify algebraic expressions and solve complex equations.
• Math tutors: Math tutors can provide you with one-on-one instruction and help you practice simplifying algebraic expressions.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By using various techniques, such as factoring, canceling, and combining like terms, we can simplify complex expressions and make them easier to solve. Remember to factor the numerator and denominator, cancel common factors, combine like terms, use the distributive property, and use the commutative property to simplify algebraic expressions.