Simplify The Expression: ( 1 2 + 1 3 ) × 8 9 \left(\frac{1}{2} + \frac{1}{3}\right) \times \frac{8}{9} ( 2 1 ​ + 3 1 ​ ) × 9 8 ​

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Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in evaluating complex expressions. In this article, we will focus on simplifying the expression (12+13)×89\left(\frac{1}{2} + \frac{1}{3}\right) \times \frac{8}{9} using basic arithmetic operations and algebraic properties.

Understanding the Expression

The given expression involves adding two fractions, 12\frac{1}{2} and 13\frac{1}{3}, and then multiplying the result by 89\frac{8}{9}. To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: None in this case.
  3. Multiplication and Division: Evaluate the multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate the addition and subtraction operations from left to right.

Adding Fractions

To add fractions, we need to have a common denominator. The least common multiple (LCM) of 2 and 3 is 6. We can rewrite the fractions with a common denominator:

12=36\frac{1}{2} = \frac{3}{6}

13=26\frac{1}{3} = \frac{2}{6}

Now, we can add the fractions:

36+26=56\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Multiplying Fractions

To multiply fractions, we simply multiply the numerators and denominators:

56×89=5×86×9=4054\frac{5}{6} \times \frac{8}{9} = \frac{5 \times 8}{6 \times 9} = \frac{40}{54}

Simplifying the Result

We can simplify the result by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 40 and 54 is 2. Dividing both the numerator and denominator by 2, we get:

4054=2027\frac{40}{54} = \frac{20}{27}

Conclusion

In this article, we simplified the expression (12+13)×89\left(\frac{1}{2} + \frac{1}{3}\right) \times \frac{8}{9} using basic arithmetic operations and algebraic properties. We followed the order of operations, added fractions with a common denominator, multiplied fractions, and simplified the result. The final simplified expression is 2027\frac{20}{27}.

Tips and Tricks for Simplifying Algebraic Expressions

  • Always follow the order of operations (PEMDAS).
  • Identify the least common multiple (LCM) of the denominators when adding fractions.
  • Multiply fractions by multiplying the numerators and denominators.
  • Simplify the result by dividing both the numerator and denominator by their greatest common divisor (GCD).

Common Mistakes to Avoid

  • Failing to follow the order of operations (PEMDAS).
  • Not identifying the least common multiple (LCM) of the denominators when adding fractions.
  • Not multiplying fractions correctly.
  • Not simplifying the result by dividing both the numerator and denominator by their greatest common divisor (GCD).

Real-World Applications of Simplifying Algebraic Expressions

Simplifying algebraic expressions has numerous real-world applications in fields such as:

  • Physics: Simplifying expressions is crucial in solving problems involving motion, energy, and forces.
  • Engineering: Simplifying expressions is essential in designing and analyzing complex systems.
  • Economics: Simplifying expressions is necessary in modeling economic systems and making predictions.

Final Thoughts

Simplifying algebraic expressions is a fundamental skill in mathematics, and it's essential to understand the steps involved in evaluating complex expressions. By following the order of operations, adding fractions with a common denominator, multiplying fractions, and simplifying the result, we can simplify even the most complex expressions. Remember to avoid common mistakes and apply the tips and tricks provided in this article to become proficient in simplifying algebraic expressions.

Introduction to the Q&A Guide

In our previous article, we provided a step-by-step guide to simplifying the expression (12+13)×89\left(\frac{1}{2} + \frac{1}{3}\right) \times \frac{8}{9}. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions, including this specific expression.

Q&A: Simplifying Algebraic Expressions

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: Evaluate any exponential expressions.
  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 add fractions with a common denominator?

A: To add fractions with a common denominator, simply add the numerators and keep the common denominator. For example:

12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Q: How do I multiply fractions?

A: To multiply fractions, simply multiply the numerators and denominators. For example:

12×34=1×32×4=38\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}

Q: How do I simplify a fraction?

A: To simplify a fraction, divide both the numerator and denominator by their greatest common divisor (GCD). For example:

1218=69=23\frac{12}{18} = \frac{6}{9} = \frac{2}{3}

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 2 and 3 is 6.

Q: How do I evaluate an expression with multiple operations?

A: To evaluate an expression with multiple operations, follow the order of operations (PEMDAS). For example:

(12+13)×89=56×89=4054=2027\left(\frac{1}{2} + \frac{1}{3}\right) \times \frac{8}{9} = \frac{5}{6} \times \frac{8}{9} = \frac{40}{54} = \frac{20}{27}

Common Mistakes to Avoid

  • Failing to follow the order of operations (PEMDAS).
  • Not identifying the least common multiple (LCM) of the denominators when adding fractions.
  • Not multiplying fractions correctly.
  • Not simplifying the result by dividing both the numerator and denominator by their greatest common divisor (GCD).

Real-World Applications of Simplifying Algebraic Expressions

Simplifying algebraic expressions has numerous real-world applications in fields such as:

  • Physics: Simplifying expressions is crucial in solving problems involving motion, energy, and forces.
  • Engineering: Simplifying expressions is essential in designing and analyzing complex systems.
  • Economics: Simplifying expressions is necessary in modeling economic systems and making predictions.

Final Thoughts

Simplifying algebraic expressions is a fundamental skill in mathematics, and it's essential to understand the steps involved in evaluating complex expressions. By following the order of operations, adding fractions with a common denominator, multiplying fractions, and simplifying the result, we can simplify even the most complex expressions. Remember to avoid common mistakes and apply the tips and tricks provided in this article to become proficient in simplifying algebraic expressions.

Additional Resources

  • Khan Academy: Algebra
  • Mathway: Algebra Calculator
  • Wolfram Alpha: Algebra Solver

Conclusion

In this article, we provided a Q&A guide to simplifying the expression (12+13)×89\left(\frac{1}{2} + \frac{1}{3}\right) \times \frac{8}{9}. We answered some of the most frequently asked questions about simplifying algebraic expressions, including this specific expression. We also provided additional resources for further learning and practice.