Simplify The Expression: $\[ \frac{5}{x} - \frac{1}{8x} \\]

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Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and mathematician should possess. In this article, we will delve into the world of algebra and explore the process of simplifying a given expression. We will focus on the expression 5x−18x\frac{5}{x} - \frac{1}{8x} and provide a step-by-step guide on how to simplify it.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at it. The expression consists of two fractions: 5x\frac{5}{x} and 18x\frac{1}{8x}. The first fraction has a numerator of 5 and a denominator of x, while the second fraction has a numerator of 1 and a denominator of 8x. Our goal is to simplify this expression by combining the two fractions.

Combining Fractions with Different Denominators

To combine fractions with different denominators, we need to find a common denominator. In this case, the least common multiple (LCM) of x and 8x is 8x. Therefore, we can rewrite the first fraction with a denominator of 8x.

5x=5×8x×8=408x\frac{5}{x} = \frac{5 \times 8}{x \times 8} = \frac{40}{8x}

Now that both fractions have the same denominator, we can subtract them.

408x−18x=40−18x=398x\frac{40}{8x} - \frac{1}{8x} = \frac{40 - 1}{8x} = \frac{39}{8x}

Simplifying the Expression

The expression 398x\frac{39}{8x} is already simplified, but we can further simplify it by canceling out any common factors between the numerator and the denominator. In this case, there are no common factors, so the expression is fully simplified.

Conclusion

Simplifying the expression 5x−18x\frac{5}{x} - \frac{1}{8x} requires a step-by-step approach. We need to find a common denominator, rewrite the fractions, and then subtract them. The final simplified expression is 398x\frac{39}{8x}. This process may seem straightforward, but it requires attention to detail and a solid understanding of algebraic manipulation.

Tips and Tricks

  • When simplifying expressions, always look for common factors between the numerator and the denominator.
  • Use the least common multiple (LCM) to find a common denominator for fractions with different denominators.
  • Rewrite fractions with a common denominator to make subtraction easier.
  • Check your work by plugging in values for the variables to ensure that the expression is simplified correctly.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

  • Physics: Simplifying expressions is essential in physics, where complex equations need to be solved to understand the behavior of physical systems.
  • Engineering: Engineers use algebraic manipulation to design and optimize systems, such as bridges and buildings.
  • Economics: Economists use algebraic manipulation to model economic systems and make predictions about future trends.

Common Mistakes to Avoid

  • Failing to find a common denominator when combining fractions with different denominators.
  • Not rewriting fractions with a common denominator before subtracting.
  • Not checking work by plugging in values for the variables.

Final Thoughts

Simplifying expressions is a fundamental aspect of algebraic manipulation. By following the steps outlined in this article, you can simplify even the most complex expressions. Remember to always look for common factors, use the least common multiple, and rewrite fractions with a common denominator. With practice and patience, you will become proficient in simplifying expressions and be able to tackle even the most challenging problems.

Additional Resources

  • Khan Academy: Algebraic Manipulation
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Algebraic Manipulation

Frequently Asked Questions

  • Q: What is the least common multiple (LCM) of two numbers? A: The LCM of two numbers is the smallest number that is a multiple of both numbers.
  • Q: How do I simplify an expression with multiple fractions? A: To simplify an expression with multiple fractions, find a common denominator and rewrite the fractions. Then, combine the fractions by adding or subtracting.
  • Q: What is the difference between a numerator and a denominator? A: The numerator is the number on top of a fraction, while the denominator is the number on the bottom.

Q&A: Simplifying Expressions and Algebraic Manipulation

Q: What is the purpose of simplifying expressions in algebra?

A: The purpose of simplifying expressions in algebra is to make complex equations easier to solve and understand. Simplifying expressions helps to reduce the complexity of an equation, making it easier to work with and analyze.

Q: How do I simplify an expression with multiple fractions?

A: To simplify an expression with multiple fractions, follow these steps:

  1. Find a common denominator for all the fractions.
  2. Rewrite each fraction with the common denominator.
  3. Combine the fractions by adding or subtracting.
  4. Simplify the resulting expression.

Q: What is the least common multiple (LCM) of two numbers?

A: The LCM of two numbers is the smallest number that is a multiple of both numbers. To find the LCM, list the multiples of each number and find the smallest number that appears in both lists.

Q: How do I simplify an expression with variables in the denominator?

A: To simplify an expression with variables in the denominator, follow these steps:

  1. Identify the variables in the denominator.
  2. Find a common denominator for all the fractions.
  3. Rewrite each fraction with the common denominator.
  4. Combine the fractions by adding or subtracting.
  5. Simplify the resulting expression.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the number on top of a fraction, while the denominator is the number on the bottom. The numerator represents the number of equal parts, while the denominator represents the total number of parts.

Q: How do I simplify an expression with negative numbers?

A: To simplify an expression with negative numbers, follow these steps:

  1. Identify the negative numbers in the expression.
  2. Rewrite the expression with positive numbers.
  3. Simplify the resulting expression.
  4. If necessary, rewrite the expression with negative numbers.

Q: What is the order of operations for simplifying expressions?

A: The order of operations for simplifying expressions is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, follow these steps:

  1. Plug in values for the variables to see if the expression simplifies correctly.
  2. Use a calculator to evaluate the expression and see if it matches the simplified expression.
  3. Check the expression for any errors or mistakes.

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Failing to find a common denominator when combining fractions with different denominators.
  • Not rewriting fractions with a common denominator before subtracting.
  • Not checking work by plugging in values for the variables.

Q: How do I simplify an expression with a variable in the numerator and a variable in the denominator?

A: To simplify an expression with a variable in the numerator and a variable in the denominator, follow these steps:

  1. Identify the variables in the numerator and denominator.
  2. Find a common denominator for all the fractions.
  3. Rewrite each fraction with the common denominator.
  4. Combine the fractions by adding or subtracting.
  5. Simplify the resulting expression.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction, while an irrational expression is an expression that cannot be written as a fraction.

Q: How do I simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, follow these steps:

  1. Identify the rational exponent.
  2. Rewrite the expression with the rational exponent.
  3. Simplify the resulting expression.

Q: What are some real-world applications of simplifying expressions?

A: Some real-world applications of simplifying expressions include:

  • Physics: Simplifying expressions is essential in physics, where complex equations need to be solved to understand the behavior of physical systems.
  • Engineering: Engineers use algebraic manipulation to design and optimize systems, such as bridges and buildings.
  • Economics: Economists use algebraic manipulation to model economic systems and make predictions about future trends.

Q: How do I use technology to simplify expressions?

A: To use technology to simplify expressions, follow these steps:

  1. Use a calculator to evaluate the expression.
  2. Use a computer algebra system (CAS) to simplify the expression.
  3. Use a graphing calculator to visualize the expression.

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include:

  • Use the least common multiple (LCM) to find a common denominator for fractions with different denominators.
  • Rewrite fractions with a common denominator to make subtraction easier.
  • Check your work by plugging in values for the variables.
  • Use a calculator or computer algebra system (CAS) to simplify expressions.