Evaluate The Expression:${ -\left{4^2-\left[5 \cdot 2^3+\sqrt{49} \cdot\left(9^2-80\right)\right]+1\right} }$

Mar 12, 2025 by ADMIN 111 views

Evaluate the Expression: A Comprehensive Guide

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Introduction

In this article, we will delve into the world of mathematical expressions and evaluate a complex expression step by step. The given expression involves various mathematical operations, including exponentiation, multiplication, addition, and subtraction. Our goal is to simplify the expression and arrive at a final value.

Understanding the Expression

The given expression is:

${ -\left\{4^2-\left[5 \cdot 2^3+\sqrt{49} \cdot\left(9^2-80\right)\right]+1\right\} }$

This expression consists of several components:

Exponentiation: $4^2$ and $2^3$
Multiplication: $5 \cdot 2^3$ and $\sqrt{49} \cdot\left(9^2-80\right)$
Addition and Subtraction: $4^2-\left[5 \cdot 2^3+\sqrt{49} \cdot\left(9^2-80\right)\right]+1$
Square Root: $\sqrt{49}$
Parentheses: Used to group and evaluate expressions

Evaluating Exponentiation

Let's start by evaluating the exponentiation components:

$4^2 = 16$
$2^3 = 8$

Evaluating Multiplication

Next, we'll evaluate the multiplication components:

$5 \cdot 2^3 = 5 \cdot 8 = 40$
$\sqrt{49} = 7$ (since $\sqrt{49} = \sqrt{7^2} = 7$ )
$\left(9^2-80\right) = 81 - 80 = 1$

Evaluating Addition and Subtraction

Now, let's evaluate the addition and subtraction components:

$5 \cdot 2^3+\sqrt{49} \cdot\left(9^2-80\right) = 40 + 7 \cdot 1 = 40 + 7 = 47$
$4^2-\left[5 \cdot 2^3+\sqrt{49} \cdot\left(9^2-80\right)\right]+1 = 16 - 47 + 1 = -30$

Final Evaluation

Finally, let's evaluate the entire expression:

${ -\left\{4^2-\left[5 \cdot 2^3+\sqrt{49} \cdot\left(9^2-80\right)\right]+1\right\} = -(-30) = 30 }$

Therefore, the final value of the given expression is 30.

Conclusion

In this article, we evaluated a complex mathematical expression step by step. We started by understanding the expression, then evaluated the exponentiation, multiplication, and addition and subtraction components. Finally, we arrived at the final value of the expression, which is 30. This guide provides a comprehensive overview of how to evaluate complex mathematical expressions and can be applied to various mathematical problems.

Frequently Asked Questions

Q: What is the order of operations in mathematics?

A: The order of operations in mathematics is:

Parentheses: Evaluate expressions inside parentheses first.
Exponentiation: Evaluate exponentiation components next.
Multiplication and Division: Evaluate multiplication and division components from left to right.
Addition and Subtraction: Finally, evaluate addition and subtraction components from left to right.

Q: How do I evaluate a complex mathematical expression?

A: To evaluate a complex mathematical expression, follow these steps:

Understand the expression and identify the components.
Evaluate the exponentiation components.
Evaluate the multiplication components.
Evaluate the addition and subtraction components.
Finally, evaluate the entire expression.

Q: What is the importance of following the order of operations?

A: Following the order of operations is crucial in mathematics because it ensures that expressions are evaluated correctly and consistently. If the order of operations is not followed, the result of an expression may be incorrect or inconsistent.

Final Thoughts

Evaluating complex mathematical expressions requires a thorough understanding of mathematical operations and the order of operations. By following the steps outlined in this guide, you can evaluate complex expressions and arrive at the correct result. Remember to always follow the order of operations and evaluate expressions step by step to ensure accuracy and consistency.

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Introduction

In our previous article, we evaluated a complex mathematical expression step by step. However, we understand that there may be many questions and doubts that readers may have. In this article, we will address some of the frequently asked questions related to evaluating mathematical expressions.

Q&A

Q: What is the order of operations in mathematics?

A: The order of operations in mathematics is:

Parentheses: Evaluate expressions inside parentheses first.
Exponentiation: Evaluate exponentiation components next.
Multiplication and Division: Evaluate multiplication and division components from left to right.
Addition and Subtraction: Finally, evaluate addition and subtraction components from left to right.

Q: How do I evaluate a complex mathematical expression?

A: To evaluate a complex mathematical expression, follow these steps:

Understand the expression and identify the components.
Evaluate the exponentiation components.
Evaluate the multiplication components.
Evaluate the addition and subtraction components.
Finally, evaluate the entire expression.

Q: What is the importance of following the order of operations?

Q: What is the difference between a mathematical expression and an equation?

A: A mathematical expression is a statement that contains variables, constants, and mathematical operations, but it does not contain an equal sign (=). An equation, on the other hand, is a statement that contains an equal sign (=) and is used to solve for a variable.

Q: How do I simplify a mathematical expression?

A: To simplify a mathematical expression, follow these steps:

Combine like terms.
Evaluate any parentheses or exponents.
Perform any multiplication or division operations.
Finally, perform any addition or subtraction operations.

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

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

Q: How do I solve a mathematical equation?

A: To solve a mathematical equation, follow these steps:

Isolate the variable on one side of the equation.
Use inverse operations to eliminate any coefficients or constants.
Finally, solve for the variable.

Q: What is the importance of checking units in mathematical expressions?

A: Checking units is crucial in mathematical expressions because it ensures that the units of the variables and constants are consistent. If the units are not consistent, the result of the expression may be incorrect or inconsistent.

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

A: To evaluate an expression with multiple variables, follow these steps:

Understand the expression and identify the variables and constants.
Evaluate any parentheses or exponents.
Perform any multiplication or division operations.
Finally, perform any addition or subtraction operations.

Conclusion

In this article, we addressed some of the frequently asked questions related to evaluating mathematical expressions. We covered topics such as the order of operations, simplifying expressions, solving equations, and checking units. By following the steps outlined in this article, you can evaluate complex mathematical expressions and arrive at the correct result.

Final Thoughts

Evaluating mathematical expressions requires a thorough understanding of mathematical operations and the order of operations. By following the steps outlined in this guide, you can evaluate complex expressions and arrive at the correct result. Remember to always follow the order of operations and evaluate expressions step by step to ensure accuracy and consistency.

Additional Resources

For more information on evaluating mathematical expressions, we recommend the following resources:

Khan Academy: Math
Mathway: Math Problem Solver
Wolfram Alpha: Math and Science Calculator

By following these resources and practicing regularly, you can improve your skills in evaluating mathematical expressions and become proficient in mathematics.