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Which of the following expressions could you rewrite using the Associative Property?(1 point)

Responses

(9⋅5)⋅(8⋅3)
left parenthesis 9 times 5 right parenthesis times left parenthesis 8 times 3 right parenthesis

(9⋅5)+8+3
left parenthesis 9 times 5 right parenthesis plus 8 plus 3

(5+8)⋅3
left parenthesis 5 plus 8 right parenthesis times 3

(5⋅8)+3

9 answers

The Associative Property states that the way numbers are grouped in addition or multiplication does not change their sum or product. Specifically, it can be applied to:

  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (a × b) × c = a × (b × c)

Now, looking at your options:

  1. (9⋅5)⋅(8⋅3): This expression involves multiplication, but the grouping is such that it contains two separate groups which cannot be rearranged to apply the Associative Property directly in its current form.

  2. (9⋅5)+8+3: This expression involves both addition and multiplication. The Associative Property can be applied to the addition part, but not as written.

  3. (5+8)⋅3: This expression involves addition grouped together with multiplication. The Associative Property isn’t applicable in a way that changes the grouping of addition within the multiplication.

  4. (5⋅8)+3: Here, it’s a multiplication first followed by addition. The Associative Property can’t be applied either.

Out of the options, none of them can be rewritten solely by using the Associative Property as they are presented since they can't group terms in a different way that adheres to the property without adding more context or operator changes. The expression that hints at potential use for the Associative Property is (5 + 8)⋅3 when considering how you might group numbers differently, but it still would not fully satisfy it as a straightforward application.

Therefore, strictly speaking, the answer is that none of these expressions can be rewritten using the Associative Property as presented.

Which of the following correctly demonstrates the Associative Property of Addition?(1 point)
Responses

x⋅(y+z)=(x⋅y)+z
x times left parenthesis y plus z right parenthesis equals left parenthesis x times y right parenthesis plus z

(x+y)+z+r=x+(y+z)+r
left parenthesis x plus y right parenthesis plus z plus r equals x plus left parenthesis y plus z right parenthesis plus r

y−(x+1)=(y−x)+1
y minus left parenthesis x plus 1 right parenthesis equals left parenthesis y minus x right parenthesis plus 1

m⋅(x⋅y)=(m⋅x)⋅y

The Associative Property of Addition states that when adding three or more numbers, the way in which the numbers are grouped does not change their sum. This can be expressed as:

\[ (a + b) + c = a + (b + c) \]

Now, let's analyze the given options:

  1. x⋅(y+z)=(x⋅y)+z: This represents the Distributive Property, not the Associative Property of Addition.

  2. (x+y)+z+r=x+(y+z)+r: This correctly illustrates the Associative Property of Addition because it shows that the grouping of the terms (x+y) can be rearranged with z and r without changing the outcome.

  3. y−(x+1)=(y−x)+1: This does not demonstrate the Associative Property of Addition, but rather a manipulation involving subtraction.

  4. m⋅(x⋅y)=(m⋅x)⋅y: This is the Associative Property of Multiplication, not Addition.

Thus, the correct answer that demonstrates the Associative Property of Addition is:

(x+y)+z+r=x+(y+z)+r.

Which of the following demonstrates the Associative Property of Multiplication?(1 point)
Responses

(3⋅5)⋅2=(3⋅5)⋅2
left parenthesis 3 times 5 right parenthesis times 2 equals left parenthesis 3 times 5 right parenthesis times 2

(3+5)⋅2=3+(5⋅2)
left parenthesis 3 plus 5 right parenthesis times 2 equals 3 plus left parenthesis 5 times 2 right parenthesis

(3⋅5)⋅2=3⋅(5⋅2)
left parenthesis 3 times 5 right parenthesis times 2 equals 3 times left parenthesis 5 times 2 right parenthesis

(3⋅5)+2=(3+5)⋅2

The Associative Property of Multiplication states that when multiplying three or more numbers, the way in which the numbers are grouped does not change their product. This can be expressed as:

\[ (a \cdot b) \cdot c = a \cdot (b \cdot c) \]

Now, let's examine the options provided:

  1. (3⋅5)⋅2=(3⋅5)⋅2: This expression simply states that something equals itself. It does not demonstrate the Associative Property at all.

  2. (3+5)⋅2=3+(5⋅2): This expression demonstrates the Distributive Property, not the Associative Property of Multiplication.

  3. (3⋅5)⋅2=3⋅(5⋅2): This correctly illustrates the Associative Property of Multiplication because it shows that the grouping of the terms can change without affecting the product.

  4. (3⋅5)+2=(3+5)⋅2: This does not demonstrate the Associative Property of Multiplication; it involves addition and multiplication in a way that is unrelated to the associative property.

Thus, the correct answer that demonstrates the Associative Property of Multiplication is:

(3⋅5)⋅2=3⋅(5⋅2).

Use the Associative Property to determine which expressions are equivalent.(1 point)
Responses

−3⋅(4x⋅−2)⋅−6y=−7xy
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals negative 7 x y

−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals left parenthesis negative 3 times 4 x right parenthesis left parenthesis negative 2 times negative 6 y right parenthesis

−3⋅(4x⋅−2)⋅−6y=18−8x
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals 18 minus 8 x

−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y

To determine which expressions are equivalent using the Associative Property, we need to examine the provided options carefully. The Associative Property states that when multiplying, the grouping of the numbers doesn’t change the product.

Let's analyze each option:

  1. −3⋅(4x⋅−2)⋅−6y = −7xy: This expression needs to be calculated to check for equivalence. When simplified, the left side would involve calculations of negatives, constants, and variables, leading to a determinate product. Without calculation, we cannot confirm equivalence.

  2. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y): This expression is valid because it demonstrates the associative property by regrouping the multiplication. This form is indeed equivalent using the associative property.

  3. −3⋅(4x⋅−2)⋅−6y = 18−8x: Similar to the first expression, this comparison needs simplification on the left to verify if it equals \( 18 - 8x \). Without performing calculations, we can't determine equivalence definitively.

  4. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)⋅(−2−6y): Here, this is incorrectly formatted, as it mixes addition with multiplication in a manner inconsistent with the multiplicative associative property.

Now focusing on correctness:

  • From options 1 and 3, we lack immediate evaluation to verify equivalency without computation.
  • Option 2 correctly demonstrates the associative property and reaffirms equivalence through regrouping.
  • Option 4 is not a correct application of the associative property for multiplication.

Thus, the expression that clearly shows equivalency using the Associative Property is:

−3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y).

According to the Associative Property, which expression is equivalent to 30m+(21m−53)+(18−2m)?(1 point)
Responses

(30m+21m)+(18−2m)
left parenthesis 30 m plus 21 m right parenthesis plus left parenthesis 18 minus 2 m right parenthesis

53m−35
53 m minus 35

51m+(−53+18)−2m
51 m plus left parenthesis negative 53 plus 18 right parenthesis minus 2 m

(49m−53)+16

To determine which expression is equivalent to \( 30m + (21m - 53) + (18 - 2m) \) according to the Associative Property, we will examine how the terms can be regrouped.

The Associative Property allows us to change the grouping of the numbers in addition without affecting the result. We will look for expressions that regroup the original expression correctly.

Let's analyze the given options:

  1. (30m + 21m) + (18 - 2m): This expression demonstrates the Associative Property by regrouping \( 30m \) and \( 21m \) together. This is a valid regrouping.

  2. 53m - 35: This expression is a simplification of \( 30m + 21m - 53 + 18 - 2m \). While it simplifies to the correct value, it does not represent a regrouping according to the Associative Property.

  3. 51m + (-53 + 18) - 2m: This expression also simplifies the original expression, but it involves rearranging the constants, which is not strictly an application of the Associative Property regarding just regrouping adds. Instead, it mixes addition and subtraction.

  4. (49m - 53) + 16: This expression appears to be a combination and doesn’t directly apply the Associative Property to the regrouping of the original expression.

So, focusing on expressions that correctly use the Associative Property, we find:

The expression (30m + 21m) + (18 - 2m) accurately demonstrates the Associative Property by regrouping the terms, and thus it is the correct answer according to the criteria you gave.