23 answers
The commutative property of addition states that the order of adding numbers does not change the sum. Therefore, you can rewrite the expression \(4x^2 + 8x\) by rearranging the terms.
One possible rearrangement could be:
\[ 8x + 4x^2 \]
Both forms represent the same expression due to the commutative property of addition.
Using the commutative property of addition, you can rearrange the terms in the expression \( ab^2 + c^3 \).
One possible rearrangement is:
\[ c^3 + ab^2 \]
Both forms represent the same expression, demonstrating the commutative property of addition.
Using the commutative property of multiplication, which states that the order of multiplying factors can be changed without affecting the product, you can rewrite the expression \( a^{20} \times x^3 \).
One possible way to rearrange it is:
\[ x^3 \times a^{20} \]
Both expressions, \( a^{20} \times x^3 \) and \( x^3 \times a^{20} \), are equivalent due to the commutative property of multiplication.
To use the commutative property to fill in the blanks so that the two algebraic expressions \( 14 __ 2 + 7 __ 14 \) are equivalent, we can think about how numbers can be arranged through addition or multiplication.
For the first expression, we can use multiplication. Thus, the first blank should be filled with "×":
For the second expression, we can also use multiplication. Thus, the second blank should be filled with "×":
Putting it together, the filled expression is:
\[ 14 \times 2 + 7 \times 14 \]
These two terms can be rearranged due to the commutative property.
I apologize for the misunderstanding. Let's correctly interpret the expressions to find equivalent algebraic forms using the commutative property of addition.
Given the expression:
To make these two expressions equivalent using the commutative property, we can fill in the blanks with addition:
The first blank should be filled with +:
The second blank should be filled with +:
So the completed expression would be:
\[ 14 + 2 + 7 + 14 \]
This maintains the equivalence through the associative property of addition as well.
If you're looking for something else, could you please clarify? Thank you!
Thank you for your patience. I see now that you're looking for a specific relationship that makes both parts of the expression equivalent, likely using the commutative property in a different way.
The correct way to interpret the expression \(14 __ 2 + 7 __ 14\) to make them equivalent would involve different operations.
Here’s how we can fill in the blanks:
For the first blank: You can fill it with the division sign \(/ \):
For the second blank: You can fill it with the multiplication sign \(\times\):
So, we would have:
\[ 14 \div 2 + 7 \times 14 \]
Evaluating both sides:
So, it seems I misunderstood your request. For them to be numerically equal (if we instead use numbers in a reasonable range), you might have wanted:
Using division in one blank and multiplication in another:
After reviewing the potential combinations and their operations, please clarify if there's a specific set of values or operations you have in mind so I can assist you correctly. Thank you, and I appreciate your help in clarifying the requirements!
The commutative property of multiplication states that you can change the order of the factors in a multiplication operation without changing the product. In other words, \(a \times b = b \times a\).
Let's evaluate each option to determine which correctly demonstrates the use of the commutative property of multiplication:
A. \(2(b^{10}+z^{11}) = (b^{10}+z^{11}) \times 2\)
B. \(z^{11} \times 2 = z^{11} + 2\)
C. \(11(b+z) = 11b + 11z\)
D. \(3(b^{10}+4) = 3(4+b^{10})\)
Given the descriptions of the options, none of the options explicitly demonstrate the commutative property of multiplication.
If I had to choose one that is closest, D is the best illustration because it involves rearranging the terms: \[3(b^{10}+4) = 3(4+b^{10})\] shows that you can change the order of the additions within the parentheses, which reflects the commutative property of addition, but not multiplication.
So overall, if you're restricted to the listed options, none truly showcase the commutative property of multiplication correctly as defined. Could you check if there might be a better option or if you'd like further clarification?
The commutative property of addition states that the order of adding numbers does not change the sum. In mathematical terms, it can be expressed as:
\[ a + b = b + a \]
Now, let's evaluate each option to identify which one correctly demonstrates the commutative property of addition:
A. \(abc + xyz = xyz + abc\)
B. \(ab + xy = zy + cba\)
C. \(0 + 579 = 579\)
D. \(123 + 456 = 579\)
The correct answer is:
A. \(abc + xyz = xyz + abc\)
This option clearly exemplifies the commutative property of addition.
To demonstrate the equivalence of the expressions \(12a + 10 + a = 10 + a + 12a\) using the commutative property, you need to rearrange the terms in the first expression.
Step 1: Use the commutative property to rearrange the terms:
The original expression is:
\[
12a + 10 + a
\]
You can rearrange it as follows using the commutative property:
\[
10 + a + 12a
\]
Now your steps would look like this:
Step 1: \(12a + 10 + a = 10 + a + 12a\) (using the commutative property of addition)
Step 2: \(10 + 13a = 10 + 13a\)
Step 3: The expressions are equivalent because they both equal the same value.
So, the missing step effectively uses the commutative property to rearrange the addition terms.
Thank you for providing more details and the options for the missing step. Let's analyze the context you've given.
You want to show how to prove the equivalence of the expressions \(12a + 10 + a\) and \(10 + a + 12a\) through logical steps, specifically using the commutative property and potentially other algebraic manipulations.
Here's how we can identify the correct missing step from your provided options:
A. \(12a + 10 = 10 + 12a\)
B. \(12a + 10 + a - a = 10 + a + 12a - a\)
C. \(10 + 12a + a = 10 + 13a\)
D. \(12a + 10 + a - 10 = 10 + a + 12a - 10\)
A. \(12a + 10 = 10 + 12a\)
This uses the commutative property to rearrange the first two terms but does not account for the \(+ a\) yet.
B. \(12a + 10 + a - a = 10 + a + 12a - a\)
This option introduces a zero effect (subtracting \(a\)), which seems unnecessarily complex for the commutative proof.
C. \(10 + 12a + a = 10 + 13a\)
This rearranges correctly and combines terms, but it's not the missing step for the immediate proof of equivalence without the prior adjustments.
D. \(12a + 10 + a - 10 = 10 + a + 12a - 10\)
This keeps the terms together while subtracting \(10\) and is not as direct as necessary.
The best choice for step 1 is likely:
A. \(12a + 10 = 10 + 12a\)
This step effectively utilizes the commutative property to rearrange part of the equation.
The complete logical flow would be:
Thank you for your clarifications!
Substituting \(1\) for \(x\) in the equation \(5x + 3 = x \times 5 + 3\) is a test case for the Commutative Property of Multiplication.
When you substitute \(x = 1\) into the equation:
Both sides equal \(8\), confirming the equivalence of the two expressions when \(x = 1\).
In summary, this substitution is a test case demonstrating the Commutative Property of Multiplication because it shows that \(5x\) can be rearranged to \(x \times 5\).
I apologize for the earlier misunderstandings in my responses. Let's clarify those questions correctly:
For the first question about demonstrating the commutative property of multiplication:
The expression \(2(b^{10} + z^{11}) = (b^{10} + z^{11}) \times 2\) does demonstrate the distributive property but also shows the commutative aspect in arrangement. I appreciate your patience and clarification.
For the third question involving the equivalence of terms:
The step \(10 + 12a + a = 10 + 13a\) does correctly illustrate combining like terms and reordering, showing the equivalence.
Thank you for your feedback, and I appreciate your understanding. If you have any more questions or need further assistance, please let me know!
