Simplify The Expression: $\frac{1}{6} - \left\{\frac{1}{2} - 3\left(\frac{1}{3} + 2\right)\right\}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. In this article, we will focus on simplifying a complex expression involving fractions and parentheses. The given expression is 16−{12−3(13+2)}\frac{1}{6} - \left\{\frac{1}{2} - 3\left(\frac{1}{3} + 2\right)\right\}. Our goal is to simplify this expression step by step, using the order of operations and basic algebraic manipulations.

Understanding the Order of Operations

Before we dive into simplifying the expression, it's essential to understand the order of operations. The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

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

Simplifying the Expression

Now that we understand the order of operations, let's simplify the given expression step by step.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is 13+2\frac{1}{3} + 2. To evaluate this expression, we need to follow the order of operations.

\frac{1}{3} + 2

Using the order of operations, we first add 2 to 13\frac{1}{3}.

\frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}

So, the expression inside the parentheses simplifies to 73\frac{7}{3}.

Step 2: Multiply 3 by the Result

Now that we have the expression inside the parentheses simplified, we can multiply 3 by the result.

3 \times \frac{7}{3} = 7

So, the expression becomes 12−7\frac{1}{2} - 7.

Step 3: Subtract 7 from 12\frac{1}{2}

Now that we have the expression simplified, we can subtract 7 from 12\frac{1}{2}.

\frac{1}{2} - 7 = -\frac{13}{2}

So, the expression becomes −132-\frac{13}{2}.

Step 4: Subtract −132-\frac{13}{2} from 16\frac{1}{6}

Finally, we can subtract −132-\frac{13}{2} from 16\frac{1}{6}.

\frac{1}{6} - \left(-\frac{13}{2}\right) = \frac{1}{6} + \frac{13}{2}

To add these fractions, we need to find a common denominator. The least common multiple of 6 and 2 is 6. So, we can rewrite the fractions with a common denominator.

\frac{1}{6} + \frac{13}{2} = \frac{1}{6} + \frac{39}{6} = \frac{40}{6} = \frac{20}{3}

So, the final simplified expression is 203\frac{20}{3}.

Conclusion

In this article, we simplified a complex expression involving fractions and parentheses using the order of operations and basic algebraic manipulations. We broke down the expression into smaller steps, evaluated each step, and finally arrived at the simplified expression. The final simplified expression is 203\frac{20}{3}. We hope this article has helped you understand how to simplify complex expressions and has provided you with a step-by-step guide to simplifying expressions.

Common Mistakes to Avoid

When simplifying expressions, it's essential to avoid common mistakes. Here are a few common mistakes to avoid:

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
  • Not evaluating expressions inside parentheses first: Evaluate expressions inside parentheses first before moving on to the next step.
  • Not finding a common denominator: Make sure to find a common denominator when adding or subtracting fractions.
  • Not simplifying fractions: Simplify fractions by dividing both the numerator and denominator by their greatest common divisor.

Practice Problems

To practice simplifying expressions, try the following problems:

  • Simplify the expression: 14−{12−2(14+1)}\frac{1}{4} - \left\{\frac{1}{2} - 2\left(\frac{1}{4} + 1\right)\right\}
  • Simplify the expression: 13+{12−3(16+2)}\frac{1}{3} + \left\{\frac{1}{2} - 3\left(\frac{1}{6} + 2\right)\right\}

References

  • "Algebra" by Michael Artin
  • "Mathematics for Dummies" by Mary Jane Sterling
  • "Simplifying Expressions" by Khan Academy

About the Author

Introduction

In our previous article, we simplified a complex expression involving fractions and parentheses using the order of operations and basic algebraic manipulations. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in simplifying expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

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

Q: Why is it essential to follow the order of operations?

A: Following the order of operations is essential because it ensures that we perform the operations in the correct order. If we don't follow the order of operations, we may get incorrect results.

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, we need to follow the order of operations. We first evaluate any exponential expressions, then multiplication and division operations, and finally addition and subtraction operations.

Q: How do I add or subtract fractions?

A: To add or subtract fractions, we need to find a common denominator. The least common multiple of the denominators is the common denominator. We then rewrite the fractions with the common denominator and add or subtract the numerators.

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

A: The numerator is the number on top of a fraction, and the denominator is the number on the bottom of a fraction.

Q: How do I simplify fractions?

A: To simplify fractions, we need to divide both the numerator and denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator without leaving a remainder.

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

A: To find the GCD of two numbers, we can use the Euclidean algorithm or list the factors of each number and find the greatest common factor.

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

A: The least common multiple (LCM) is the smallest number that is a multiple of both numbers.

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

A: To find the LCM of two numbers, we can list the multiples of each number and find the smallest common multiple.

Practice Problems

To practice simplifying expressions, try the following problems:

  • Simplify the expression: 14−{12−2(14+1)}\frac{1}{4} - \left\{\frac{1}{2} - 2\left(\frac{1}{4} + 1\right)\right\}
  • Simplify the expression: 13+{12−3(16+2)}\frac{1}{3} + \left\{\frac{1}{2} - 3\left(\frac{1}{6} + 2\right)\right\}
  • Simplify the expression: 12−{13−2(16+1)}\frac{1}{2} - \left\{\frac{1}{3} - 2\left(\frac{1}{6} + 1\right)\right\}

References

  • "Algebra" by Michael Artin
  • "Mathematics for Dummies" by Mary Jane Sterling
  • "Simplifying Expressions" by Khan Academy

About the Author

The author of this article is a mathematics educator with over 10 years of experience teaching algebra and other math subjects. The author has a passion for making math accessible and fun for students of all ages.

Additional Resources

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