Question

The following is a proof of the algebraic equivalency of (2x)³ and 8x³. Fill in each of the blanks with either the statement commutative property or associative property.

(2x)³ =2x∙2x∙2x
=2(x×2)(x×2)x ___________________
=2(2x)(2x)x ___________________
=2∙2(x×2)x∙x ___________________
=2∙2(2x)x∙x ___________________
=(2∙2∙2)(x∙x∙x) ___________________
=8x³

Answer 1

You're using the commutative property when you swap the order of factors in a multiplication:

`a\cdot b = b\cdot a`

and you use the associative property when you regroup the products of more than 2 factors in a different way:

`(a\cdot b)\cdot c = a\cdot (b\cdot c)`

So, you're constantly alternating between associative and commutative. Try to see which property you're using in the first step, and then keep alternating between the two!

Answer 2

The reasons for each statement are show below.

Explanation:

We need to prove (2x)³ is equivalent to 8x³.

Commutative property: According to the commutative property of multiplication

`a\cdot b=b\cdot a`

Associative property: According to the associative property of multiplication

`a\cdot (b\cdot c)=(a\cdot b)\cdot c`

The given expression (2x)³ can be written as

`(2x)^3=(2x)\cdot (2x)\cdot (2x)`

`(2x)^3=2(x\cdot 2)(x\cdot 2)x` (Associative property)

`(2x)^3=2(2x)(2x)x` (Commutative property)

`(2x)^3=(2\cdot 2)(x\cdot 2)(x\cdot x)` (Associative property)

`(2x)^3=(2\cdot 2)(2\cdot x)(x\cdot x)` (Commutative property)

`(2x)^3=(2\cdot 2\cdot 2)(x\cdot x\cdot x)` (Associative property)

`(2x)^3=8x^3`

Hence proved.