Why Is the Distributive Property Key in Simplifying Algebraic Expressions?

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SUMMARY

The discussion centers on the application of the distributive property in simplifying algebraic expressions involving an unknown real number, denoted as x. The participants analyze three specific expressions: (2x)x=2x², (x+3)x=x²+3x, and 4(x+3)=4x+4×3, confirming that the distributive property justifies the latter two, while the associative property of multiplication is required for the first expression. The importance of defining x as an unknown real number and the notation of squaring are emphasized to ensure clarity in algebraic operations.

PREREQUISITES
  • Understanding of the distributive property in algebra.
  • Familiarity with the associative property of multiplication.
  • Basic knowledge of real number axioms.
  • Ability to interpret algebraic expressions and operations.
NEXT STEPS
  • Study the properties of real numbers, focusing on the distributive and associative properties.
  • Learn about the axioms of algebra, particularly those relevant to real numbers.
  • Practice simplifying algebraic expressions using the distributive property.
  • Explore the implications of notation in algebra, especially regarding operations on variables.
USEFUL FOR

Students learning algebra, educators teaching algebraic concepts, and anyone seeking to deepen their understanding of algebraic properties and expressions.

bergausstein
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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!
 
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bergausstein said:
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.
 
what axiom do i need for a? let me guess, is it an axiom of equality?
 
bergausstein said:
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...
 
I like Serena said:
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?
 
bergausstein said:
we can use associative property of multiplication (ab)b = a(bb) am i correct?

Yep!
 

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