Why Is the Distributive Property Key in Simplifying Algebraic Expressions?

[bergausstein]

in the following exercises, assume that x stands for an unknown real number, and assume that $x^2=x\times x$. which of the properties of real numbers justifies each of the following statement?

a. $(2x)x=2x^2$
b. $(x+3)x=x^2+3x$
c. $4(x+3)=4x+4\times 3$

my answers
a. distributive property
b. distributive property
c. distributive property.

i just don't know if my answers are complete. and I'm also bothered why this part " assume that x stands for an unknown real number, and assume that $x^2=x\times x$." is important.

thanks!

 

[I like Serena]

in the following exercises, assume that x stands for an unknown real number, and assume that $x^2=x\times x$. which of the properties of real numbers justifies each of the following statement?

a. $(2x)x=2x^2$
b. $(x+3)x=x^2+3x$
c. $4(x+3)=4x+4*3$

my answers
a. distributive property
b. distributive property
c. distributive property.

i just don't know if my answers are complete. and I'm also bothered why this part " assume that x stands for an unknown real number, and assume that $x^2=x\times x$." is important.

thanks!

Hi bergausstein!

The extra assumptions are needed, since in abstract algebra you can't really assume anything. In principle you're limited to exactly what the axioms give you. Anything else needs to be specified. These assumptions are matters of notation, so that you know that squaring a number is in all respects the same as multiplying that number by itself.
Actually, the extra assumptions in this case are so standard, that I consider it a bit of overkill to mention them.

Your answers to (b) and (c) are correct. However, for (a) you will need a different axiom.

Btw, is there a reason you used a different multiplication operator in (c)?
Luckily there is only 1 multiplication operator in the field of the real numbers, but otherwise that would be ambiguous.

 

[bergausstein]

what axiom do i need for a? let me guess, is it an axiom of equality?

 

[I like Serena]

what axiom do i need for a? let me guess, is it an axiom of equality?

Axiom of equality? That's not really one of the axioms of the real numbers, although you are implicitly using the axioms that belong to an equivalence relation.

What I mean is that the distributive property is a(b+c)=ab+ac.
But if I look at (a) I have (2x)x = 2(xx).
Those do not look like they have the same structure...

 

[bergausstein]

Axiom of equality? That's not really one of the axioms of the real numbers, although you are implicitly using the axioms that belong to an equivalence relation.

What I mean is that the distributive property is a(b+c)=ab+ac.
But if I look at (a) I have (2x)x = 2(xx).
Those do not look like they have the same structure...

we can use associative property of multiplication (ab)b = a(bb) am i correct?

 

[I like Serena]

we can use associative property of multiplication (ab)b = a(bb) am i correct?

Yep!