Calculate The Expression: $\[ 8 \cdot (1 + 2)^2 \\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them requires a clear understanding of the rules and operations involved. In this article, we will focus on calculating the expression ${ 8 \cdot (1 + 2)^2 }$ and provide a step-by-step guide on how to solve it.

Understanding the Expression

The given expression is ${ 8 \cdot (1 + 2)^2 }$. To solve this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: Evaluate any exponents (such as squaring or cubing).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is ${ 1 + 2 }$. To evaluate this expression, we simply add 1 and 2:

1+2=3{ 1 + 2 = 3 }

So, the expression inside the parentheses is equal to 3.

Step 2: Square the Result

Now that we have evaluated the expression inside the parentheses, we need to square the result. The expression to be squared is ${ 3^2 }$. To square a number, we multiply it by itself:

32=3×3=9{ 3^2 = 3 \times 3 = 9 }

So, the result of squaring the expression inside the parentheses is 9.

Step 3: Multiply 8 by the Result

Now that we have squared the result, we need to multiply 8 by the result. The expression to be multiplied is ${ 8 \times 9 }$. To multiply two numbers, we simply multiply them together:

8×9=72{ 8 \times 9 = 72 }

So, the final result of the expression is 72.

Conclusion

In this article, we have solved the expression ${ 8 \cdot (1 + 2)^2 }$ using the order of operations (PEMDAS). We first evaluated the expression inside the parentheses, then squared the result, and finally multiplied 8 by the result. The final result of the expression is 72.

Tips and Tricks

Here are some tips and tricks to help you solve algebraic expressions:

  • Always follow the order of operations (PEMDAS).
  • Evaluate expressions inside parentheses first.
  • Square exponents (such as squaring or cubing) next.
  • Multiply and divide from left to right.
  • Add and subtract from left to right.

Common Mistakes

Here are some common mistakes to avoid when solving algebraic expressions:

  • Not following the order of operations (PEMDAS).
  • Evaluating expressions inside parentheses incorrectly.
  • Squaring exponents incorrectly.
  • Multiplying and dividing incorrectly.
  • Adding and subtracting incorrectly.

Real-World Applications

Algebraic expressions have many real-world applications, including:

  • Science: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of populations.
  • Engineering: Algebraic expressions are used to design and optimize systems, such as bridges and buildings.
  • Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.

Conclusion

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.

Q: What are the rules for solving algebraic expressions?

A: The rules for solving algebraic expressions are based on the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: Evaluate any exponents (such as squaring or cubing).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, you need to follow the order of operations (PEMDAS). First, evaluate any exponents (such as squaring or cubing), then multiply and divide from left to right, and finally add and subtract 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. A constant is a value that does not change.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms. Like terms are terms that have the same variable and exponent.

Q: What is the distributive property?

A: The distributive property is a rule that allows you to multiply a single term by multiple terms. It is written as:

a(b + c) = ab + ac

Q: How do I use the distributive property to simplify an algebraic expression?

A: To use the distributive property to simplify an algebraic expression, you need to multiply each term inside the parentheses by the term outside the parentheses.

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

A: An equation is a statement that says two expressions are equal. An expression is a mathematical statement that contains variables, constants, and mathematical operations.

Q: How do I solve an equation?

A: To solve an equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

A: The order of operations (PEMDAS) is a rule that tells you which operations to perform first when evaluating an algebraic expression. It is written as:

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: Evaluate any exponents (such as squaring or cubing).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: Why is it important to follow the order of operations (PEMDAS)?

A: Following the order of operations (PEMDAS) is important because it ensures that you evaluate expressions correctly and avoid errors.

Q: Can I use a calculator to solve algebraic expressions?

A: Yes, you can use a calculator to solve algebraic expressions. However, it is still important to understand the rules and operations involved in solving algebraic expressions.

Q: How do I practice solving algebraic expressions?

A: You can practice solving algebraic expressions by working through problems in a textbook or online resource. You can also try solving algebraic expressions on your own and then checking your answers with a calculator or online resource.