Evaluate The Following Expression:${ \frac{3(2+x)+4(3+4x)}{12+16x} }$ { \frac{6+3x}{18+19x} \}

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Introduction

In mathematics, evaluating complex algebraic expressions is a crucial skill that requires a deep understanding of mathematical concepts and techniques. These expressions often involve multiple variables, operations, and functions, making them challenging to simplify and solve. In this article, we will evaluate the given expression and provide a step-by-step guide on how to simplify it.

The Given Expression

The given expression is:

3(2+x)+4(3+4x)12+16x\frac{3(2+x)+4(3+4x)}{12+16x}

6+3x18+19x\frac{6+3x}{18+19x}

Step 1: Simplify the Numerator

To simplify the given expression, we will start by simplifying the numerator. We can do this by applying the distributive property to each term in the numerator.

3(2+x)=6+3x3(2+x) = 6 + 3x

4(3+4x)=12+16x4(3+4x) = 12 + 16x

Now, we can combine like terms in the numerator:

6+3x+12+16x=18+19x6 + 3x + 12 + 16x = 18 + 19x

Step 2: Simplify the Denominator

The denominator is already simplified, so we can move on to the next step.

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator.

18+19x18+19x\frac{18 + 19x}{18 + 19x}

Step 4: Cancel Out Common Factors

We can cancel out the common factor of 18+19x18 + 19x in the numerator and denominator.

11\frac{1}{1}

Conclusion

In this article, we evaluated the given expression and simplified it step by step. We applied the distributive property to simplify the numerator, simplified the denominator, and then simplified the expression by dividing the numerator by the denominator. Finally, we canceled out the common factor in the numerator and denominator to get the final simplified expression.

Real-World Applications

Evaluating complex algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, algebraic expressions are used to describe the motion of objects and the behavior of physical systems. In engineering, algebraic expressions are used to design and optimize complex systems, such as bridges and buildings. In economics, algebraic expressions are used to model economic systems and make predictions about future economic trends.

Tips and Tricks

When evaluating complex algebraic expressions, it's essential to follow these tips and tricks:

  • Simplify the numerator and denominator separately: This will make it easier to simplify the expression as a whole.
  • Apply the distributive property: This will help you simplify the numerator and denominator by breaking down complex expressions into simpler ones.
  • Cancel out common factors: This will help you simplify the expression by eliminating common factors in the numerator and denominator.
  • Check your work: This will help you ensure that your simplified expression is correct and accurate.

Common Mistakes

When evaluating complex algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not simplifying the numerator and denominator separately: This can make it difficult to simplify the expression as a whole.
  • Not applying the distributive property: This can make it difficult to simplify the numerator and denominator.
  • Not canceling out common factors: This can make it difficult to simplify the expression.
  • Not checking your work: This can lead to errors and inaccuracies in your simplified expression.

Conclusion

Evaluating complex algebraic expressions is a crucial skill that requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article, you can simplify complex expressions and make them easier to understand and work with. Remember to simplify the numerator and denominator separately, apply the distributive property, cancel out common factors, and check your work to ensure accuracy and accuracy.

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Introduction

In our previous article, we evaluated a complex algebraic expression and provided a step-by-step guide on how to simplify it. In this article, we will answer some frequently asked questions about evaluating complex algebraic expressions.

Q: What is a complex algebraic expression?

A complex algebraic expression is an expression that involves multiple variables, operations, and functions. It can be a combination of simple expressions, such as numbers, variables, and constants, and more complex expressions, such as fractions, exponents, and roots.

Q: How do I simplify a complex algebraic expression?

To simplify a complex algebraic expression, you can follow these steps:

  1. Simplify the numerator and denominator separately: This will make it easier to simplify the expression as a whole.
  2. Apply the distributive property: This will help you simplify the numerator and denominator by breaking down complex expressions into simpler ones.
  3. Cancel out common factors: This will help you simplify the expression by eliminating common factors in the numerator and denominator.
  4. Check your work: This will help you ensure that your simplified expression is correct and accurate.

Q: What is the distributive property?

The distributive property is a mathematical property that states that for any numbers a, b, and c:

a(b + c) = ab + ac

This property can be applied to simplify complex expressions by breaking down the expression into simpler ones.

Q: How do I apply the distributive property?

To apply the distributive property, you can follow these steps:

  1. Identify the expression to be simplified: This can be a complex expression involving multiple variables, operations, and functions.
  2. Apply the distributive property: This involves breaking down the expression into simpler ones by multiplying each term in the expression by the other terms.
  3. Simplify the resulting expression: This involves combining like terms and eliminating any common factors.

Q: What is a common factor?

A common factor is a term that appears in both the numerator and denominator of an expression. It can be a number, a variable, or a combination of both.

Q: How do I cancel out common factors?

To cancel out common factors, you can follow these steps:

  1. Identify the common factor: This can be a number, a variable, or a combination of both.
  2. Cancel out the common factor: This involves eliminating the common factor from both the numerator and denominator.
  3. Simplify the resulting expression: This involves combining like terms and eliminating any remaining common factors.

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

Some common mistakes to avoid when evaluating complex algebraic expressions include:

  • Not simplifying the numerator and denominator separately: This can make it difficult to simplify the expression as a whole.
  • Not applying the distributive property: This can make it difficult to simplify the numerator and denominator.
  • Not canceling out common factors: This can make it difficult to simplify the expression.
  • Not checking your work: This can lead to errors and inaccuracies in your simplified expression.

Q: How can I practice evaluating complex algebraic expressions?

You can practice evaluating complex algebraic expressions by:

  • Working through examples: This can help you understand the steps involved in simplifying complex expressions.
  • Solving problems: This can help you apply the concepts and techniques learned in a practical way.
  • Using online resources: This can provide you with additional practice and support.

Conclusion

Evaluating complex algebraic expressions is a crucial skill that requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article, you can simplify complex expressions and make them easier to understand and work with. Remember to simplify the numerator and denominator separately, apply the distributive property, cancel out common factors, and check your work to ensure accuracy and accuracy.