Evaluate The Expression: $(4n+4)^2$ [Provide Your Answer In The Box Below]$\square$ Submit

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Understanding the Problem

In this problem, we are tasked with evaluating the expression (4n+4)2(4n+4)^2. To do this, we need to apply the rules of algebra and follow the order of operations to simplify the expression.

The Order of Operations

The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. 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 any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Evaluating the Expression

Now that we understand the order of operations, let's apply it to the expression (4n+4)2(4n+4)^2.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is 4n+44n+4. We can simplify this expression by combining like terms:

4n+4=4(n+1)4n+4 = 4(n+1)

Step 2: Apply the Exponent

Now that we have simplified the expression inside the parentheses, we can apply the exponent:

(4(n+1))2=(4(n+1))(4(n+1))(4(n+1))^2 = (4(n+1))(4(n+1))

Step 3: Multiply the Terms

To multiply the terms, we need to follow the distributive property:

(4(n+1))(4(n+1))=4(4(n+1))(n+1)(4(n+1))(4(n+1)) = 4(4(n+1))(n+1)

Step 4: Simplify the Expression

Now that we have multiplied the terms, we can simplify the expression:

4(4(n+1))(n+1)=16(n+1)24(4(n+1))(n+1) = 16(n+1)^2

Step 5: Expand the Squared Term

To expand the squared term, we need to apply the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2:

(n+1)2=n2+2n+1(n+1)^2 = n^2 + 2n + 1

Step 6: Substitute the Expanded Term

Now that we have expanded the squared term, we can substitute it back into the expression:

16(n+1)2=16(n2+2n+1)16(n+1)^2 = 16(n^2 + 2n + 1)

Step 7: Distribute the 16

To distribute the 16, we need to multiply each term by 16:

16(n2+2n+1)=16n2+32n+1616(n^2 + 2n + 1) = 16n^2 + 32n + 16

The Final Answer

The final answer is:

16n^2 + 32n + 16$<br/> # **Evaluating Algebraic Expressions: A Step-by-Step Guide** =========================================================== ## **Understanding the Problem** ----------------------------- In this problem, we are tasked with evaluating the expression $(4n+4)^2$. To do this, we need to apply the rules of algebra and follow the order of operations to simplify the expression. ## **The Order of Operations** --------------------------- The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations: 1. **P**arentheses: Evaluate expressions inside parentheses first. 2. **E**xponents: Evaluate any exponential expressions next. 3. **M**ultiplication and **D**ivision: Evaluate any multiplication and division operations from left to right. 4. **A**ddition and **S**ubtraction: Finally, evaluate any addition and subtraction operations from left to right. ## **Evaluating the Expression** --------------------------- Now that we understand the order of operations, let's apply it to the expression $(4n+4)^2$. ### **Step 1: Evaluate the Expression Inside the Parentheses** --------------------------------------------------------- The expression inside the parentheses is $4n+4$. We can simplify this expression by combining like terms: $4n+4 = 4(n+1)

Step 2: Apply the Exponent

Now that we have simplified the expression inside the parentheses, we can apply the exponent:

(4(n+1))2=(4(n+1))(4(n+1))(4(n+1))^2 = (4(n+1))(4(n+1))

Step 3: Multiply the Terms

To multiply the terms, we need to follow the distributive property:

(4(n+1))(4(n+1))=4(4(n+1))(n+1)(4(n+1))(4(n+1)) = 4(4(n+1))(n+1)

Step 4: Simplify the Expression

Now that we have multiplied the terms, we can simplify the expression:

4(4(n+1))(n+1)=16(n+1)24(4(n+1))(n+1) = 16(n+1)^2

Step 5: Expand the Squared Term

To expand the squared term, we need to apply the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2:

(n+1)2=n2+2n+1(n+1)^2 = n^2 + 2n + 1

Step 6: Substitute the Expanded Term

Now that we have expanded the squared term, we can substitute it back into the expression:

16(n+1)2=16(n2+2n+1)16(n+1)^2 = 16(n^2 + 2n + 1)

Step 7: Distribute the 16

To distribute the 16, we need to multiply each term by 16:

16(n2+2n+1)=16n2+32n+1616(n^2 + 2n + 1) = 16n^2 + 32n + 16

The Final Answer

The final answer is:

16n2+32n+1616n^2 + 32n + 16

Frequently Asked Questions

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. 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 any 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 an expression with parentheses?

A: To simplify an expression with parentheses, you need to evaluate the expression inside the parentheses first. This means that you need to follow the order of operations and evaluate any exponential expressions, multiplication and division operations, and addition and subtraction operations inside the parentheses.

Q: How do I apply an exponent to an expression?

A: To apply an exponent to an expression, you need to multiply the expression by itself as many times as the exponent indicates. For example, if you have the expression (x+1)2(x+1)^2, you need to multiply the expression (x+1)(x+1) by itself twice.

Q: How do I distribute a coefficient to a term?

A: To distribute a coefficient to a term, you need to multiply the coefficient by each term in the expression. For example, if you have the expression 2(x+1)2(x+1), you need to multiply the coefficient 2 by each term in the expression, which gives you 2x+22x+2.

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

A: A coefficient is a number that is multiplied by a variable or a term, while a constant is a number that is not multiplied by a variable or a term. For example, in the expression 2x2x, the 2 is a coefficient because it is multiplied by the variable x, while the 3 in the expression 3x+23x+2 is a constant because it is not multiplied by a variable or a term.

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

A: To evaluate an expression with multiple operations, you need to follow the order of operations and evaluate each operation in the correct order. This means that you need to evaluate any exponential expressions, multiplication and division operations, and addition and subtraction operations from left to right.

Conclusion

Evaluating algebraic expressions can be a challenging task, but by following the order of operations and simplifying expressions step by step, you can arrive at the correct answer. Remember to always follow the order of operations and simplify expressions step by step to ensure that you arrive at the correct answer.