Which Equation Shows An Example Of The Associative Property Of Addition?A. ( − 4 + I ) + 4 I = − 4 + ( I + 4 I (-4+i)+4i=-4+(i+4i ( − 4 + I ) + 4 I = − 4 + ( I + 4 I ] B. ( − 4 + I ) + 4 I = 4 I + ( − 4 I + I (-4+i)+4i=4i+(-4i+i ( − 4 + I ) + 4 I = 4 I + ( − 4 I + I ] C. 4 I × ( − 4 I + I ) = ( 4 I − 4 I ) + ( 4 I × I 4i \times(-4i+i)=(4i-4i)+(4i \times I 4 I × ( − 4 I + I ) = ( 4 I − 4 I ) + ( 4 I × I ] D. ( − 4 I + I ) + 0 = ( − 4 I + I (-4i+i)+0=(-4i+i ( − 4 I + I ) + 0 = ( − 4 I + I ]

The associative property of addition is a fundamental concept in mathematics that states that the order in which we add numbers does not change the result. In other words, when we have three numbers, say a, b, and c, we can add them in any order and still get the same result. This property is denoted by the equation (a + b) + c = a + (b + c).

In this article, we will explore which of the given equations shows an example of the associative property of addition.

What is the Associative Property of Addition?

The associative property of addition is a mathematical property that states that the order in which we add numbers does not change the result. This property is often denoted by the equation (a + b) + c = a + (b + c), where a, b, and c are any numbers.

To understand this property, let's consider an example. Suppose we have three numbers: 2, 3, and 4. We can add them in any order and still get the same result. For instance, if we add 2 and 3 first, we get 5, and then add 4 to get 9. On the other hand, if we add 3 and 4 first, we get 7, and then add 2 to get 9. As we can see, the order in which we add the numbers does not change the result.

Examples of the Associative Property of Addition

Now that we have a basic understanding of the associative property of addition, let's look at some examples. We will examine each of the given equations and determine which one shows an example of the associative property of addition.

Equation A: $(-4+i)+4i=-4+(i+4i)$

Let's start by examining Equation A. This equation states that $(-4+i)+4i=-4+(i+4i)$. To determine if this equation shows an example of the associative property of addition, let's simplify both sides of the equation.

On the left-hand side, we have $(-4+i)+4i$. Using the distributive property of addition, we can rewrite this as $-4+4i+i$. Combining like terms, we get $-4+5i$.

On the right-hand side, we have $-4+(i+4i)$. Using the distributive property of addition, we can rewrite this as $-4+i+4i$. Combining like terms, we get $-4+5i$.

As we can see, both sides of the equation simplify to $-4+5i$. Therefore, Equation A shows an example of the associative property of addition.

Equation B: $(-4+i)+4i=4i+(-4i+i)$

Now let's examine Equation B. This equation states that $(-4+i)+4i=4i+(-4i+i)$. To determine if this equation shows an example of the associative property of addition, let's simplify both sides of the equation.

On the left-hand side, we have $(-4+i)+4i$. Using the distributive property of addition, we can rewrite this as $-4+4i+i$. Combining like terms, we get $-4+5i$.

On the right-hand side, we have $4i+(-4i+i)$. Using the distributive property of addition, we can rewrite this as $4i-4i+i$. Combining like terms, we get $-4+5i$.

As we can see, both sides of the equation simplify to $-4+5i$. Therefore, Equation B shows an example of the associative property of addition.

Equation C: $4i \times(-4i+i)=(4i-4i)+(4i \times i)$

Now let's examine Equation C. This equation states that $4i \times(-4i+i)=(4i-4i)+(4i \times i)$. To determine if this equation shows an example of the associative property of addition, let's simplify both sides of the equation.

On the left-hand side, we have $4i \times(-4i+i)$. Using the distributive property of multiplication, we can rewrite this as $4i \times (-4i) + 4i \times i$. Combining like terms, we get $-16i^2 + 4i^2$. Since $i^2 = -1$, we can simplify this to $-16(-1) + 4(-1)$. This simplifies to $16 - 4$, which is equal to $12$.

On the right-hand side, we have $(4i-4i)+(4i \times i)$. Using the distributive property of addition, we can rewrite this as $0 + 4i \times i$. Combining like terms, we get $4i \times i$. Since $i^2 = -1$, we can simplify this to $4(-1)$, which is equal to $-4$.

As we can see, the left-hand side of the equation simplifies to $12$, while the right-hand side simplifies to $-4$. Therefore, Equation C does not show an example of the associative property of addition.

Equation D: $(-4i+i)+0=(-4i+i)$

Now let's examine Equation D. This equation states that $(-4i+i)+0=(-4i+i)$. To determine if this equation shows an example of the associative property of addition, let's simplify both sides of the equation.

On the left-hand side, we have $(-4i+i)+0$. Using the distributive property of addition, we can rewrite this as $-4i + i + 0$. Combining like terms, we get $-4i + i$, which is equal to $-3i$.

On the right-hand side, we have $-4i+i$. Combining like terms, we get $-3i$.

As we can see, both sides of the equation simplify to $-3i$. Therefore, Equation D shows an example of the associative property of addition.

Conclusion

In conclusion, we have examined four equations and determined which one shows an example of the associative property of addition. We found that Equations A, B, and D all show examples of the associative property of addition, while Equation C does not.

The associative property of addition is a fundamental concept in mathematics that states that the order in which we add numbers does not change the result. This property is denoted by the equation (a + b) + c = a + (b + c), where a, b, and c are any numbers.

The associative property of addition is a fundamental concept in mathematics that states that the order in which we add numbers does not change the result. In this article, we will answer some frequently asked questions about the associative property of addition.

Q: What is the associative property of addition?

A: The associative property of addition is a mathematical property that states that the order in which we add numbers does not change the result. This property is often denoted by the equation (a + b) + c = a + (b + c), where a, b, and c are any numbers.

Q: Why is the associative property of addition important?

A: The associative property of addition is important because it allows us to add numbers in any order and still get the same result. This property is used in many mathematical operations, such as solving equations and inequalities.

Q: How do I apply the associative property of addition?

A: To apply the associative property of addition, simply add the numbers in any order and still get the same result. For example, if we have the equation (2 + 3) + 4, we can add the numbers in any order and still get the same result: 2 + (3 + 4) = 2 + 7 = 9.

Q: Can I use the associative property of addition with negative numbers?

A: Yes, you can use the associative property of addition with negative numbers. For example, if we have the equation (-2 + 3) + 4, we can add the numbers in any order and still get the same result: -2 + (3 + 4) = -2 + 7 = 5.

Q: Can I use the associative property of addition with fractions?

A: Yes, you can use the associative property of addition with fractions. For example, if we have the equation (1/2 + 1/4) + 3/4, we can add the numbers in any order and still get the same result: 1/2 + (1/4 + 3/4) = 1/2 + 1 = 3/2.

Q: Can I use the associative property of addition with decimals?

A: Yes, you can use the associative property of addition with decimals. For example, if we have the equation (2.5 + 1.2) + 3.8, we can add the numbers in any order and still get the same result: 2.5 + (1.2 + 3.8) = 2.5 + 5 = 7.5.

Q: Can I use the associative property of addition with imaginary numbers?

A: Yes, you can use the associative property of addition with imaginary numbers. For example, if we have the equation (2i + 3i) + 4i, we can add the numbers in any order and still get the same result: 2i + (3i + 4i) = 2i + 7i = 9i.

Q: Can I use the associative property of addition with complex numbers?

A: Yes, you can use the associative property of addition with complex numbers. For example, if we have the equation (2 + 3i) + (4 - 5i), we can add the numbers in any order and still get the same result: 2 + (3i + 4 - 5i) = 2 + (-2 + 4) = 4.

Conclusion

In conclusion, the associative property of addition is a fundamental concept in mathematics that states that the order in which we add numbers does not change the result. We hope that this article has provided a clear understanding of the associative property of addition and has helped to answer some frequently asked questions about this property.