Donte Simplified The Expression Below:${ \begin{array}{l} 4(1+3 I)-(8-5 I) \ 4+3 I-8+5 I \ -4+8 I \end{array} }$What Mistake Did Donte Make?A. He Did Not Apply The Distributive Property Correctly For ${ 4(1+3 I)\$} .B. He Did Not

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Understanding the Distributive Property

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The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. It is essential to apply the distributive property correctly to simplify complex expressions. In this article, we will explore a common mistake made by Donte in simplifying an algebraic expression and discuss the correct application of the distributive property.

The Expression to Simplify

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The expression given by Donte is:

$$\begin{array}{l} 4(1+3 i)-(8-5 i) \\ 4+3 i-8+5 i \\ -4+8 i \end{array}$$

Donte's Mistake

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Donte's mistake lies in the application of the distributive property. Let's break down the expression step by step to identify the error.

Step 1: Apply the Distributive Property

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To simplify the expression, we need to apply the distributive property to the first term, $4(1+3 i)$. This means we multiply the term outside the parentheses, 4, with each term inside the parentheses, 1 and $3 i$.

$$4(1+3 i) = 4(1) + 4(3 i)$$

Using the distributive property, we get:

$$4(1+3 i) = 4 + 12 i$$

Step 2: Subtract the Second Term

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Now, we need to subtract the second term, $(8-5 i)$, from the result obtained in Step 1.

$$4 + 12 i - (8 - 5 i)$$

To subtract the second term, we need to distribute the negative sign to each term inside the parentheses.

$$4 + 12 i - 8 + 5 i$$

Step 3: Combine Like Terms

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Now, we can combine like terms to simplify the expression.

$$4 - 8 + 12 i + 5 i$$

Combining like terms, we get:

$$-4 + 17 i$$

Correct Application of the Distributive Property

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The correct application of the distributive property would result in the following expression:

$$4(1+3 i) = 4 + 12 i$$

Subtracting the second term, $(8-5 i)$, from the result, we get:

$$4 + 12 i - 8 + 5 i$$

Combining like terms, we get:

$$-4 + 17 i$$

Conclusion

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In conclusion, Donte's mistake was not applying the distributive property correctly for the term $4(1+3 i)$. By distributing the term outside the parentheses with each term inside, we can simplify the expression correctly. The correct application of the distributive property is essential in algebra to simplify complex expressions and solve equations.

Common Mistakes in Algebra

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Mistake 1: Not Applying the Distributive Property

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One common mistake in algebra is not applying the distributive property correctly. This can lead to incorrect simplification of expressions and solutions to equations.

Mistake 2: Not Combining Like Terms

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Another common mistake is not combining like terms. This can result in incorrect solutions to equations and expressions.

Mistake 3: Not Following the Order of Operations

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The order of operations is a fundamental concept in algebra that dictates the order in which we perform mathematical operations. Not following the order of operations can lead to incorrect solutions and expressions.

Tips for Avoiding Common Mistakes

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Tip 1: Read the Problem Carefully

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Before starting to solve a problem, read it carefully to understand what is being asked. This will help you avoid common mistakes and ensure that you are solving the correct problem.

Tip 2: Apply the Distributive Property Correctly

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The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. Apply the distributive property correctly to simplify complex expressions.

Tip 3: Combine Like Terms

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Combining like terms is essential in algebra to simplify expressions and solve equations. Make sure to combine like terms correctly to avoid common mistakes.

Conclusion

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In conclusion, the distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. By applying the distributive property correctly, we can simplify complex expressions and solve equations. Common mistakes in algebra include not applying the distributive property, not combining like terms, and not following the order of operations. By following the tips outlined in this article, we can avoid common mistakes and ensure that we are solving problems correctly.

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Frequently Asked Questions

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Q: What is the distributive property?

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A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside.

Q: How do I apply the distributive property?

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A: To apply the distributive property, multiply the term outside the parentheses with each term inside the parentheses. For example, if we have the expression $4(1+3 i)$, we would multiply 4 with each term inside the parentheses, resulting in $4 + 12 i$.

Q: What is the difference between the distributive property and the commutative property?

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A: The distributive property and the commutative property are two different concepts in algebra. The distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside, while the commutative property states that the order of the terms in an expression does not change the result.

Q: Can I apply the distributive property to expressions with more than one term inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with more than one term inside the parentheses. For example, if we have the expression $4(1+3 i+2)$, we would multiply 4 with each term inside the parentheses, resulting in $4 + 12 i + 8$.

Q: What is the difference between the distributive property and the associative property?

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A: The distributive property and the associative property are two different concepts in algebra. The distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside, while the associative property states that the order in which we perform operations does not change the result.

Q: Can I apply the distributive property to expressions with variables inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with variables inside the parentheses. For example, if we have the expression $4(x+3)$, we would multiply 4 with each term inside the parentheses, resulting in $4x + 12$.

Q: What is the importance of the distributive property in algebra?

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A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. This property is essential in solving equations and simplifying expressions.

Q: Can I apply the distributive property to expressions with fractions inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with fractions inside the parentheses. For example, if we have the expression $4(\frac{1}{2} + \frac{1}{3})$, we would multiply 4 with each term inside the parentheses, resulting in $2 + \frac{4}{3}$.

Q: What is the difference between the distributive property and the identity property?

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A: The distributive property and the identity property are two different concepts in algebra. The distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside, while the identity property states that the product of a number and 1 is the number itself.

Q: Can I apply the distributive property to expressions with exponents inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with exponents inside the parentheses. For example, if we have the expression $4(x^2 + 3)$, we would multiply 4 with each term inside the parentheses, resulting in $4x^2 + 12$.

Q: What is the importance of the distributive property in real-life applications?

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A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. This property is essential in solving equations and simplifying expressions, which is crucial in real-life applications such as finance, engineering, and science.

Q: Can I apply the distributive property to expressions with negative numbers inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with negative numbers inside the parentheses. For example, if we have the expression $4(-2 + 3)$, we would multiply 4 with each term inside the parentheses, resulting in $-8 + 12$.

Q: What is the difference between the distributive property and the inverse property?

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A: The distributive property and the inverse property are two different concepts in algebra. The distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside, while the inverse property states that the product of a number and its reciprocal is 1.

Q: Can I apply the distributive property to expressions with decimals inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with decimals inside the parentheses. For example, if we have the expression $4(2.5 + 3.2)$, we would multiply 4 with each term inside the parentheses, resulting in $10 + 12.8$.

Q: What is the importance of the distributive property in solving equations?

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A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. This property is essential in solving equations, as it allows us to simplify expressions and isolate variables.

Q: Can I apply the distributive property to expressions with mixed numbers inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with mixed numbers inside the parentheses. For example, if we have the expression $4(2\frac{1}{2} + 3)$, we would multiply 4 with each term inside the parentheses, resulting in $10 + 12$.

Q: What is the difference between the distributive property and the distributive property of multiplication over addition?

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A: The distributive property and the distributive property of multiplication over addition are two different concepts in algebra. The distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside, while the distributive property of multiplication over addition states that the product of a number and the sum of two numbers is equal to the sum of the products of the number with each of the two numbers.

Q: Can I apply the distributive property to expressions with imaginary numbers inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with imaginary numbers inside the parentheses. For example, if we have the expression $4(2 + 3i)$, we would multiply 4 with each term inside the parentheses, resulting in $8 + 12i$.

Q: What is the importance of the distributive property in simplifying expressions?

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A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. This property is essential in simplifying expressions, as it allows us to combine like terms and eliminate unnecessary parentheses.

Q: Can I apply the distributive property to expressions with radical expressions inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with radical expressions inside the parentheses. For example, if we have the expression $4(\sqrt{2} + 3)$, we would multiply 4 with each term inside the parentheses, resulting in $4\sqrt{2} + 12$.

Q: What is the difference between the distributive property and the distributive property of multiplication over subtraction?

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A: The distributive property and the distributive property of multiplication over subtraction are two different concepts in algebra. The distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside, while the distributive property of multiplication over subtraction states that the product of a number and the difference of two numbers is equal to the difference of the products of the number with each of the two numbers.

Q: Can I apply the distributive property to expressions with absolute value expressions inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with absolute value expressions inside the parentheses. For example, if we have the expression $4(|x| + 3)$, we would multiply 4 with each term inside the parentheses, resulting in $4|x| + 12$.

Q: What is the importance of the distributive property in solving systems of equations?

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A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. This property is essential in solving systems of equations, as it allows us to simplify expressions and isolate variables.

Q: Can I apply the distributive property to expressions with rational expressions inside the parentheses?

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A: Yes, you can apply the distributive property to expressions with rational expressions inside the parentheses. For example, if we have the expression $4(\frac{x}{y} + 3)$, we would multiply 4 with each term inside the parentheses, resulting in $\frac{4x}{y} + 12$.

Q: What is the difference between the distributive property and the distributive property of multiplication over division?

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A: The distributive property and the distributive property of multiplication over division are two different concepts in algebra. The distributive property allows us to expand expressions by multiplying each term inside the parentheses