Simplify The Expression:${ \frac{-9-23}{(-2)^3} }$

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Introduction

Mathematical expressions can be complex and daunting, but with a clear understanding of the rules and operations involved, they can be simplified and evaluated with ease. In this article, we will focus on simplifying the expression −9−23(−2)3\frac{-9-23}{(-2)^3}, breaking down the steps involved and providing a clear explanation of each operation.

Understanding the Expression

The given expression is a fraction, consisting of a numerator and a denominator. The numerator is the sum of two negative numbers, -9 and -23, while the denominator is the cube of -2. To simplify this expression, we need to follow the order of operations (PEMDAS), which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Evaluating the Numerator

The numerator of the expression is -9 + (-23). To evaluate this, we need to follow the order of operations, which means we need to perform the addition first. However, since both numbers are negative, we can combine them by adding their absolute values and then applying the negative sign.

numerator = -9 + (-23)
numerator = -(9 + 23)
numerator = -32

Evaluating the Denominator

The denominator of the expression is (-2)^3. To evaluate this, we need to raise -2 to the power of 3. This means we need to multiply -2 by itself three times.

denominator = (-2)^3
denominator = (-2) * (-2) * (-2)
denominator = -8

Simplifying the Expression

Now that we have evaluated the numerator and the denominator, we can simplify the expression by dividing the numerator by the denominator.

simplified_expression = numerator / denominator
simplified_expression = -32 / -8
simplified_expression = 4

Conclusion

In conclusion, simplifying the expression −9−23(−2)3\frac{-9-23}{(-2)^3} involves evaluating the numerator and the denominator separately and then dividing the numerator by the denominator. By following the order of operations and performing the necessary calculations, we can simplify the expression and arrive at the final answer.

Frequently Asked Questions

  • What is the order of operations? The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
  • How do I evaluate a fraction? To evaluate a fraction, you need to divide the numerator by the denominator. If the numerator is greater than the denominator, the result will be a positive number. If the numerator is less than the denominator, the result will be a negative number.
  • What is the difference between a numerator and a denominator? The numerator is the top number in a fraction, while the denominator is the bottom number. The numerator is the number being divided, while the denominator is the number by which we are dividing.

Additional Resources

  • Khan Academy: Order of Operations
  • Mathway: Simplifying Fractions
  • Wolfram Alpha: Evaluating Expressions

Step-by-Step Guide

  1. Evaluate the numerator by adding -9 and -23.
  2. Evaluate the denominator by raising -2 to the power of 3.
  3. Simplify the expression by dividing the numerator by the denominator.
  4. Check the final answer to ensure it is correct.

Common Mistakes

  • Failing to follow the order of operations
  • Not evaluating the numerator and denominator separately
  • Not simplifying the expression correctly

Tips and Tricks

  • Make sure to follow the order of operations carefully
  • Evaluate the numerator and denominator separately before simplifying the expression
  • Check the final answer to ensure it is correct

Real-World Applications

  • Simplifying expressions is a crucial skill in mathematics and is used in a variety of real-world applications, such as finance, engineering, and science.
  • Understanding the order of operations is essential for evaluating complex mathematical expressions and making accurate calculations.
  • Simplifying fractions is a fundamental concept in mathematics and is used in a variety of real-world applications, such as cooking, building, and finance.

Introduction

Mathematical expressions can be complex and daunting, but with a clear understanding of the rules and operations involved, they can be simplified and evaluated with ease. In this article, we will focus on simplifying the expression −9−23(−2)3\frac{-9-23}{(-2)^3}, breaking down the steps involved and providing a clear explanation of each operation.

Q&A: Simplifying Expressions

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I evaluate a fraction?

A: To evaluate a fraction, you need to divide the numerator by the denominator. If the numerator is greater than the denominator, the result will be a positive number. If the numerator is less than the denominator, the result will be a negative number.

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

A: The numerator is the top number in a fraction, while the denominator is the bottom number. The numerator is the number being divided, while the denominator is the number by which we are dividing.

Q: How do I simplify a complex expression?

A: To simplify a complex expression, you need to follow the order of operations, which means you need to perform the operations in the following order:

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponents (such as squaring or cubing).
  3. Perform any multiplication and division operations from left to right.
  4. Perform any addition and subtraction operations from left to right.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change.

Q: How do I evaluate an expression with variables?

A: To evaluate an expression with variables, you need to substitute the value of the variable into the expression and then perform the necessary operations.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation by performing the necessary operations.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a.

Q&A: Real-World Applications

Q: How is simplifying expressions used in real-world applications?

A: Simplifying expressions is used in a variety of real-world applications, such as finance, engineering, and science. For example, in finance, simplifying expressions is used to calculate interest rates and investment returns. In engineering, simplifying expressions is used to design and optimize systems. In science, simplifying expressions is used to model and analyze complex systems.

Q: How is the order of operations used in real-world applications?

A: The order of operations is used in a variety of real-world applications, such as finance, engineering, and science. For example, in finance, the order of operations is used to calculate interest rates and investment returns. In engineering, the order of operations is used to design and optimize systems. In science, the order of operations is used to model and analyze complex systems.

Q&A: Tips and Tricks

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include:

  • Following the order of operations carefully
  • Evaluating the numerator and denominator separately before simplifying the expression
  • Checking the final answer to ensure it is correct

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

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

  • Failing to follow the order of operations
  • Not evaluating the numerator and denominator separately
  • Not simplifying the expression correctly

Conclusion

In conclusion, simplifying expressions is a crucial skill in mathematics and is used in a variety of real-world applications. By following the order of operations and performing the necessary calculations, we can simplify complex expressions and arrive at the final answer. Remember to follow the tips and tricks outlined in this article to avoid common mistakes and ensure accurate results.

Frequently Asked Questions

  • What is the order of operations?
  • How do I evaluate a fraction?
  • What is the difference between a numerator and a denominator?
  • How do I simplify a complex expression?
  • What is the difference between a variable and a constant?
  • How do I evaluate an expression with variables?
  • What is the difference between a linear equation and a quadratic equation?
  • How do I solve a linear equation?
  • How do I solve a quadratic equation?

Additional Resources

  • Khan Academy: Order of Operations
  • Mathway: Simplifying Fractions
  • Wolfram Alpha: Evaluating Expressions
  • Khan Academy: Linear Equations
  • Khan Academy: Quadratic Equations
  • Mathway: Solving Linear Equations
  • Mathway: Solving Quadratic Equations