2 x (3x7)=(2x3) x 7.   6+3=3+6.   7x2=2x7.    18x1=18.   26-(9-7) doesn't equal (26-9)-7. The question is: What MATH PROPERTIES have been applied to show that the two expressions are...

2 x (3x7)=(2x3) x 7.   6+3=3+6.   7x2=2x7.    18x1=18.   26-(9-7) doesn't equal (26-9)-7.

The question is: What MATH PROPERTIES have been applied to show that the two expressions are equivalent or not equivalent?

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

(a) 2x(3x7)=(2x3)x7 is an example of the associative property of multiplication. (Think of the people you associate with as your group -- the associative property for addition or multiplication allows you to change where the grouping symbols are placed.)

(b) 6+3=3+6 is an example of the commutative property of addition.

(c) 7x2=2x7 is an example of the commutative property of multiplication.

(d) 18x1=18 is an example of the multiplicative identity property.

(e) 26-(9-7) `ne ` (26-9)-7 since there is no associative property of subtraction.

Note that there is no associative property for subtraction or division, nor is there a commutative property of subtraction nor division. In order to use these properties, all operations must be converted to addition and multiplication.

We change subtraction to addition by adding the opposite (additive inverse) while we change division to multiplication by multiplying by the reciprocal (multiplicative inverse.)

So by the order of operations 26-9-7=(26-9)-7=17-7=10.(Subtract in order from left to right.) In order to write an equivalent expression, we rewrite the original as a sum:

26 +(-9)+(-7). Now the associative property of addition can be applied:

26+((-9)+(-7))=(26+(-9))+(-7)

26+(-16)=17+(-7)

10=10

shumbm's profile pic

Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

Posted on

Hello!

The properties, or laws, that can be applied here, are (`a,` `b,` `c` denote any integer, real or complex numbers):

1a) The Commutative Law of Addition:

`a+b=b+a`

1b) The Commutative Law of Multiplication:

`a*b=b*a`

 

2a) The Associative Law of Addition:

`(a+b)+c=a+(b+c)`

2b) The Associative Law of Multiplication:

`(a*b)*c=a*(b*c)`

 

3) The Distributive Law (of Multiplication with respect to Addition):

`a*(b+c)=a*b+a*c`

 

Now consider our specific problems.

1. 2 x (3x7) = (2x3) x 7.
Here the associative law of multiplication (2b) is applied (with `a=2,` `b=3` and `c=7`).

2. 6+3=3+6.
This is the commutative law of addition (1a), `a=6` and `b=3.`

3. 7x2=2x7.
This is the commutative law of multiplication (1b), `a=7` and `b=2.`

4. 18x1=18.
This property of multiplication, `a*1=a,` may be considered as the definition of multiplication by `1.`

 

 

` `

Problem 5. `26-(9-7) != (26-9)-7.`

Here we see an attempt to apply hypothetical "associative law of subtraction". But there is NO such law! To apply the associative law of addition, we have to represent subtraction as addition of additive inverse:

`a-b=a+(-1)*b.`

Then `26-(9-7) = 26+(-1)*(9-7) = 26+(-1)*[9+(-1)*7].`

Here we can apply the distributive law and obtain

`26-(9-7) = 26+[(-1)*9+(-1)*((-1)*7)].`

Now use associative law of multiplication to get that `(-1)*((-1)*7) = ((-1)*(-1))*7 = 1*7=7.`

So the initial expression is equal to `26+[(-1)*9+7],`
which is equal to `(26+(-1)*9)+7` by the associative law of addition,
which in turn is equal to `(26-9)+7.`

Sources:
rmgsh4's profile pic

rmgsh4 | Elementary School Teacher | In Training Educator

Posted on

In the simplest terms possible:

2 x (3x7)=(2x3) x 7 This is an example of the Associative Property of Multiplication. We know this because only the parenthesis moved, the numbers stayed the same and no matter how you multiply these numbers together the answer will be the same. Think of it like people and who they associate with at school. Three girls are friends, one day Jill might hang out with Betty and the next day she might hang out with Sue but all three of them are still friends! 

6+3=3+6 This is an example of the Commutative Property of Addition. The numbers simply changed places, in other words they "commuted" but the answer is still the same no matter how you add them.

7x2=2x7 This is an example of the Commutative Property of Multiplication. Again, the numbers simply changed places, they "commuted" but the answer is still the same no matter how you multiply them.

18x1=18 This is an example of the Multiplicative Identity Property. That property simply states that anything multiplied by 1 is itself. Simple as that! 

26-(9-7) doesn't equal (26-9)-7 Now this one is not an example of any property. I think this is an attempt to possibly show an associative property of subtraction but that is not a real thing! There is no Associative or Commutative property of subtraction or division. In order to make this a valid statement you would have to convert the subtraction to addition by applying integer rules.

And if you were to convert to addition and apply integer rules, you would see that both sets of numbers equal 10, therefore the whole statement is false or incorrect because 10=10

Hope that helps and explains it in a way that is easy enough to understand! :)

Good Luck!

 

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