Simplify: 8 - 2 (14 - 10 ÷ 2)² ÷ 3⁴

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Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. The given expression, 8 - 2 (14 - 10 ÷ 2)² ÷ 3⁴, is a perfect example of a complex expression that requires careful simplification. In this article, we will break down the expression step by step and simplify it using the order of operations (PEMDAS).

Understanding the Order of Operations

Before we dive into simplifying the expression, it's essential to understand the order of operations, which 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 expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next (e.g., 2²).
  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.

Breaking Down the Expression

Now that we understand the order of operations, let's break down the given expression:

8 - 2 (14 - 10 ÷ 2)² ÷ 3⁴

We can see that there are multiple operations involved, including parentheses, exponents, multiplication, division, and subtraction. To simplify the expression, we need to follow the order of operations.

Step 1: Evaluate Expressions Inside Parentheses

The first step is to evaluate the expression inside the innermost parentheses: 14 - 10 ÷ 2.

14 - 10 ÷ 2 = 14 - 5 = 9

Step 2: Evaluate Exponential Expressions

Next, we need to evaluate the exponential expression: (14 - 10 ÷ 2)².

(14 - 10 ÷ 2)² = 9² = 81

Step 3: Multiply and Divide

Now, we need to multiply and divide the expression: 2 × 81 ÷ 3⁴.

2 × 81 = 162
3⁴ = 81
162 ÷ 81 = 2

Step 4: Subtract

Finally, we need to subtract the result from the original expression: 8 - 2.

8 - 2 = 6

Conclusion

By following the order of operations, we have simplified the complex expression: 8 - 2 (14 - 10 ÷ 2)² ÷ 3⁴.

The final answer is: 6

Tips and Tricks

  • When simplifying complex expressions, it's essential to follow the order of operations (PEMDAS).
  • Use parentheses to group expressions and make it easier to evaluate.
  • Exponential expressions should be evaluated before multiplication and division.
  • Finally, addition and subtraction operations should be evaluated from left to right.

Practice Problems

  • Simplify the expression: 3 × (2 + 5)² ÷ 4
  • Simplify the expression: 10 - 2 (7 - 3)² ÷ 5
  • Simplify the expression: 6 × (4 - 2)² ÷ 3

Real-World Applications

Simplifying complex expressions is a crucial skill in mathematics that has many real-world applications. For example:

  • In physics, simplifying complex expressions helps us solve problems related to motion, energy, and momentum.
  • In engineering, simplifying complex expressions helps us design and optimize systems, such as electrical circuits and mechanical systems.
  • In finance, simplifying complex expressions helps us calculate interest rates, investment returns, and risk management.

Conclusion

In conclusion, simplifying complex expressions is a crucial skill in mathematics that requires careful attention to the order of operations. By following the order of operations (PEMDAS), we can simplify even the most complex expressions and arrive at the correct solution.

Introduction

In our previous article, we simplified the complex expression: 8 - 2 (14 - 10 ÷ 2)² ÷ 3⁴. In this article, we will answer some frequently asked questions related to simplifying complex expressions.

Q&A

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 expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next (e.g., 2²).
  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: How do I simplify complex expressions?

A: To simplify complex expressions, follow the order of operations (PEMDAS):

  1. Evaluate expressions inside parentheses first.
  2. Evaluate any exponential expressions next.
  3. Evaluate multiplication and division operations from left to right.
  4. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between multiplication and division?

A: Multiplication and division are both operations that involve numbers, but they have different properties. Multiplication is a commutative operation, meaning that the order of the numbers does not change the result (e.g., 2 × 3 = 3 × 2). Division, on the other hand, is a non-commutative operation, meaning that the order of the numbers does change the result (e.g., 6 ÷ 2 ≠ 2 ÷ 6).

Q: How do I handle negative numbers in complex expressions?

A: When simplifying complex expressions with negative numbers, remember to follow the order of operations (PEMDAS). If a negative number is inside parentheses, evaluate the expression inside the parentheses first. If a negative number is outside parentheses, evaluate the expression outside the parentheses first.

Q: Can I simplify complex expressions with fractions?

A: Yes, you can simplify complex expressions with fractions. To simplify fractions, follow the order of operations (PEMDAS) and simplify the numerator and denominator separately.

Q: How do I simplify complex expressions with decimals?

A: To simplify complex expressions with decimals, follow the order of operations (PEMDAS) and simplify the expression as you would with integers.

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

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

  • Not following the order of operations (PEMDAS)
  • Not evaluating expressions inside parentheses first
  • Not evaluating exponential expressions next
  • Not evaluating multiplication and division operations from left to right
  • Not evaluating addition and subtraction operations from left to right

Tips and Tricks

  • When simplifying complex expressions, it's essential to follow the order of operations (PEMDAS).
  • Use parentheses to group expressions and make it easier to evaluate.
  • Exponential expressions be evaluated before multiplication and division.
  • Finally, addition and subtraction operations should be evaluated from left to right.

Practice Problems

  • Simplify the expression: 3 × (2 + 5)² ÷ 4
  • Simplify the expression: 10 - 2 (7 - 3)² ÷ 5
  • Simplify the expression: 6 × (4 - 2)² ÷ 3

Real-World Applications

Simplifying complex expressions is a crucial skill in mathematics that has many real-world applications. For example:

  • In physics, simplifying complex expressions helps us solve problems related to motion, energy, and momentum.
  • In engineering, simplifying complex expressions helps us design and optimize systems, such as electrical circuits and mechanical systems.
  • In finance, simplifying complex expressions helps us calculate interest rates, investment returns, and risk management.

Conclusion

In conclusion, simplifying complex expressions is a crucial skill in mathematics that requires careful attention to the order of operations (PEMDAS). By following the order of operations and avoiding common mistakes, we can simplify even the most complex expressions and arrive at the correct solution.