Simplify The Expression: ${ -\frac{8}{11} - \left(-\frac{5}{11}\right) }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. It involves combining like terms, removing unnecessary operations, and rearranging the expression to make it easier to understand. In this article, we will focus on simplifying the expression ${-\frac{8}{11} - \left(-\frac{5}{11}\right)}$. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding the Expression

The given expression is ${-\frac{8}{11} - \left(-\frac{5}{11}\right)}$. This expression involves two fractions, ${-\frac{8}{11}}$ and ${-\frac{5}{11}}$. The negative sign in front of each fraction indicates that they are both negative numbers.

Step 1: Remove the Negative Signs

When we see a negative sign in front of a fraction, it means that the fraction is negative. However, when we see a negative sign inside a fraction, it means that the fraction is positive. In this case, we have two negative signs, one in front of each fraction and one inside each fraction. To simplify the expression, we need to remove the negative signs.

-\frac{8}{11} - (-\frac{5}{11}) = \frac{8}{11} + \frac{5}{11}

Step 2: Combine the Fractions

Now that we have removed the negative signs, we can combine the fractions. To do this, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the denominators are 11 and 11, so the LCM is 11.

\frac{8}{11} + \frac{5}{11} = \frac{8+5}{11} = \frac{13}{11}

Step 3: Simplify the Expression

Now that we have combined the fractions, we can simplify the expression. In this case, the expression is already simplified, so we don't need to do anything else.

Conclusion

Simplifying the expression ${-\frac{8}{11} - \left(-\frac{5}{11}\right)}$ involves removing the negative signs and combining the fractions. By following these steps, we can simplify the expression and arrive at the final answer, which is 1311\frac{13}{11}.

Real-World Applications

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. For example, in physics, we often need to simplify complex expressions to solve problems involving motion, energy, and momentum. In engineering, we need to simplify expressions to design and optimize systems, such as electrical circuits and mechanical systems.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

  • Always remove the negative signs before combining fractions.
  • Find the least common multiple (LCM) of the denominators to combine fractions.
  • Simplify the expression by combining like terms and removing unnecessary operations.
  • Use a calculator or a computer algebra system (CAS) to check your work and ensure that your answer is correct.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions:

  • Failing to remove the negative signs before combining fractions.
  • Not finding the least common multiple (LCM) of the denominators.
  • Not simplifying the expression by combining like terms and removing unnecessary operations.
  • Not checking your work with a calculator or a computer algebra system (CAS).

Conclusion

Introduction

In our previous article, we discussed how to simplify the expression ${-\frac{8}{11} - \left(-\frac{5}{11}\right)}$. In this article, we will provide a Q&A guide to help you understand the concept of simplifying expressions and address any questions you may have.

Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to make them easier to understand and work with. By simplifying expressions, we can combine like terms, remove unnecessary operations, and arrive at a more manageable solution.

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

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

  1. Remove the negative signs before combining fractions.
  2. Find the least common multiple (LCM) of the denominators.
  3. Combine the fractions by adding or subtracting the numerators.
  4. Simplify the expression by combining like terms and removing unnecessary operations.

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

A: The least common multiple (LCM) of two numbers is the smallest number that both numbers can divide into evenly. For example, the LCM of 4 and 6 is 12, because both 4 and 6 can divide into 12 evenly.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can follow these steps:

  1. List the multiples of each number.
  2. Identify the smallest number that appears in both lists.
  3. The smallest number that appears in both lists is the LCM.

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

A: The numerator is the top number in a fraction, and the denominator is the bottom number. For example, in the fraction 34\frac{3}{4}, the numerator is 3 and the denominator is 4.

Q: How do I simplify an expression with variables?

A: To simplify an expression with variables, you need to follow these steps:

  1. Combine like terms by adding or subtracting the coefficients.
  2. Simplify the expression by removing unnecessary operations.
  3. Use the distributive property to expand the expression.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows you to expand an expression by multiplying each term by a factor. For example, the distributive property states that a(b+c)=ab+aca(b+c) = ab + ac.

Q: How do I use the distributive property to simplify an expression?

A: To use the distributive property to simplify an expression, you need to follow these steps:

  1. Identify the terms in the expression.
  2. Multiply each term by the factor.
  3. Combine like terms by adding or subtracting the coefficients.

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

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

  • Failing to remove the negative signs before combining fractions.
  • Not finding the least common multiple (LCM) of the denominators.
  • Not simplifying the expression by combining like terms and removing unnecessary operations.
  • Not checking your work with a calculator or a computer algebra system (CAS).

Conclusion

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can simplify expressions and arrive at a more manageable solution. Remember to always remove the negative signs before combining fractions, find the least common multiple (LCM) of the denominators, and simplify the expression by combining like terms and removing unnecessary operations.