Simplify The Expression \[$(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)\$\].

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When faced with a complex expression like $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$, it's essential to break it down into manageable parts and follow the order of operations. In this article, we'll guide you through the process of simplifying this expression, step by step.

Understanding the Order of Operations

Before we dive into simplifying the expression, let's review the order of operations, which is a set of rules that helps us evaluate mathematical expressions in the correct order. 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've reviewed the order of operations, let's apply it to the given expression:

$(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$

Step 1: Evaluate Expressions Inside Parentheses

The first step is to evaluate the expressions inside the parentheses. We have two sets of parentheses: $(2 \cdot 5 - 3)$ and $\left(3 - 7^2\right)$. Let's start with the first set:

$(2 \cdot 5 - 3)$

To evaluate this expression, we need to follow the order of operations:

1. Multiply 2 and 5: $2 \cdot 5 = 10$
2. Subtract 3 from the result: $10 - 3 = 7$

So, the first set of parentheses evaluates to 7.

Step 2: Evaluate Exponential Expressions

Next, we need to evaluate the exponential expression inside the second set of parentheses:

$\left(3 - 7^2\right)$

To evaluate this expression, we need to follow the order of operations:

1. Evaluate the exponent: $7^2 = 49$
2. Subtract 3 from the result: $49 - 3 = 46$

So, the second set of parentheses evaluates to 46.

Step 3: Simplify the Expression

Now that we've evaluated the expressions inside the parentheses, we can simplify the original expression:

$(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$

$= 7^2 - 46$

$= 49 - 46$

$= 3$

And that's it! The simplified expression is 3.

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us solve problems efficiently and accurately. By following the order of operations and breaking down complex expressions into manageable parts, we can simplify even the most challenging expressions. In this article, we've guided you through the process of simplifying the expression $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$, step by step. With practice and patience, you'll become proficient in simplifying expressions and tackling even the most complex mathematical problems.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is a set of rules that helps us evaluate mathematical expressions in the correct order. The acronym PEMDAS is commonly used to remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Q: How do I simplify a complex expression? A: To simplify a complex expression, follow the order of operations and break down the expression into manageable parts. Evaluate expressions inside parentheses first, then evaluate any exponential expressions, and finally, evaluate any multiplication and division operations from left to right.
• Q: What is the simplified expression for $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$? A: The simplified expression is 3.

Additional Resources

• For more information on the order of operations, visit the Khan Academy website.
• For practice problems and exercises, visit the Mathway website.
• For a comprehensive guide to simplifying expressions, visit the Purplemath website.
  

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Introduction

In our previous article, we guided you through the process of simplifying the expression $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$, step by step. However, we know that sometimes, it's not enough to just read through a tutorial or guide. You may have questions, and that's where our Q&A guide comes in. In this article, we'll answer some of the most frequently asked questions about simplifying expressions, including the expression $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$.

Q&A Guide

Q: What is the order of operations?

A: The order of operations is a set of rules that helps us evaluate mathematical expressions in the correct order. The acronym PEMDAS is commonly used to remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I simplify a complex expression?

A: To simplify a complex expression, follow the order of operations and break down the expression into manageable parts. Evaluate expressions inside parentheses first, then evaluate any exponential expressions, and finally, evaluate any multiplication and division operations from left to right.

Q: What is the difference between a parenthesis and a bracket?

A: A parenthesis and a bracket are both used to group expressions together, but they are used in different contexts. A parenthesis is used to group expressions that are not part of the main expression, while a bracket is used to group expressions that are part of the main expression.

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, follow the order of operations. First, evaluate any expressions inside the parentheses, then evaluate the exponent, and finally, multiply the result by the base.

Q: What is the simplified expression for $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$?

A: The simplified expression is 3.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions, but it's not always the best option. Calculators can be prone to errors, and they may not always follow the order of operations correctly. It's always best to simplify expressions by hand, using the order of operations.

Q: How do I know if an expression is simplified?

A: An expression is simplified when it can be evaluated using the order of operations, and the result is a single value. If an expression contains any variables or unknown values, it is not simplified.

Q: Can I simplify expressions with variables?

A: Yes, you can simplify expressions with variables, but it's a bit more complicated. When simplifying expressions with variables, you need to follow the order of operations, and you need to be careful not to introduce any new variables or unknown values.

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us solve problems efficiently and accurately. By following the order of operations and breaking down complex expressions into manageable parts, we can simplify even the most challenging expressions. In this Q&A guide, we've answered some of the most frequently asked questions about simplifying expressions, including the expression $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$. With practice and patience, you'll become proficient in simplifying expressions and tackling even the most complex mathematical problems.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is a set of rules that helps us evaluate mathematical expressions in the correct order. The acronym PEMDAS is commonly used to remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Q: How do I simplify a complex expression? A: To simplify a complex expression, follow the order of operations and break down the expression into manageable parts. Evaluate expressions inside parentheses first, then evaluate any exponential expressions, and finally, evaluate any multiplication and division operations from left to right.
• Q: What is the simplified expression for $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$? A: The simplified expression is 3.

Additional Resources

• For more information on the order of operations, visit the Khan Academy website.
• For practice problems and exercises, visit the Mathway website.
• For a comprehensive guide to simplifying expressions, visit the Purplemath website.

Related Articles

• Simplify the Expression: A Step-by-Step Guide to Evaluating $(2 \cdot 5 - 3)^2 - \left(3 - 7^2\right)$
• The Order of Operations: A Guide to Evaluating Mathematical Expressions
• Simplifying Expressions with Variables: A Guide to Evaluating Complex Expressions