Why Is the Distributive Property Key in Simplifying Algebraic Expressions?

In summary: That's correct. The associative property of multiplication justifies the statement in (a). Great job! In summary, the properties of real numbers used to justify the statements in the given exercises are the distributive property (a, b, c) and the associative property (a).
  • #1
bergausstein
191
0
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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  • #2
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.
 
  • #3
what axiom do i need for a? let me guess, is it an axiom of equality?
 
  • #4
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...
 
  • #5
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?
 
  • #6
bergausstein said:
we can use associative property of multiplication (ab)b = a(bb) am i correct?

Yep!
 

FAQ: Why Is the Distributive Property Key in Simplifying Algebraic Expressions?

What is the definition of the additive inverse property?

The additive inverse property states that for any real number a, there exists a real number b such that a + b = 0. In other words, every real number has an opposite that, when added together, equal zero.

How does the multiplicative inverse property relate to division?

The multiplicative inverse property states that for any non-zero real number a, there exists a real number b such that a * b = 1. This means that the inverse of a number is its reciprocal, and when multiplying by the reciprocal, the result is always 1. This is directly related to division, as dividing by a number is the same as multiplying by its reciprocal.

What is the commutative property of addition and multiplication?

The commutative property of addition states that for any two real numbers a and b, a + b = b + a. This means that the order in which we add two numbers does not change the result. The commutative property of multiplication follows the same idea, stating that for any two real numbers a and b, a * b = b * a.

How does the associative property differ between addition and multiplication?

The associative property of addition states that for any three real numbers a, b, and c, (a + b) + c = a + (b + c). This means that when adding three or more numbers, the grouping does not affect the result. On the other hand, the associative property of multiplication states that for any three real numbers a, b, and c, (a * b) * c = a * (b * c). This means that when multiplying three or more numbers, the grouping does not affect the result.

How do the distributive properties apply to real numbers?

The distributive property of multiplication over addition states that for any three real numbers a, b, and c, a * (b + c) = (a * b) + (a * c). This means that when multiplying a number by a sum, we can distribute the multiplication to each term and then add the results. The distributive property of multiplication over subtraction follows the same idea, stating that for any three real numbers a, b, and c, a * (b - c) = (a * b) - (a * c).

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