I am having trouble understanding the difference between expressions such as (x^6) * (x^4) and (x^6)^4.   Can you help me understand the difference?Can you explain this in more general terms for...

I am having trouble understanding the difference between expressions such as (x^6) * (x^4) and (x^6)^4.   Can you help me understand the difference?

Can you explain this in more general terms for me also?

Asked on by gabejohnson

3 Answers | Add Yours

hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

(x^6)*(x^4)

That means we will multiply x by itself 6 times and then multiply it by x bet itself 4 times:

OR : (x*x*x*x*x*x) = x^6

        (x*x*x*x) = x^4

==> (x^4)*(x^6) = (x*x*x*x)*(x*x*x*x*x*x) = x^10

Then we conclude that:

(x^4)*(x^6) = x^(4+6) = x^10

 

On the other hand:

(x^6)^4  means that (x^6) multiply by itsef 4 times:

==> (x^6)^4 = (x^6)*(x^6)*(x^6)*(x^6)

= (x*x*x*x*x*x)*(x*x*x*x*x*x)*(x*x*x*x*x*x)*(x*x*x*x*x*x)

= x^24

Then we concude thatL

(x^6)^4 = x^(6*4) = x^24

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

In general, there are 2 rules:

The first rule: if we are multiplying 2 power functions, with matching bases, the exponents are added.

In this case, we have (x^6) * (x^4). The matching base is x and the exponents are 6 and 4.

(x^6) * (x^4) = x^(6+4) = x^10

We could also write x^6 as 6 times x.

x^6 = x*x*x*x*x*x

This is also the case in which the bases are matching and we'll calculate the sum of exponents:

x^6 = x^(1+1+1+1+1+1) = x^6

We'll do the same with x^4 = x^(1+1+1+1)

The second rule: If we'll raise a power to another power, the exponents are multiplied.

(x^6)^4 = x^(6*4) = x^24

Now, let's treat the problem in this way:

If x^6 = x^(1+1+1+1+1+1)

(x^6)^4 = x^6*x^6*x^6*x^6 = x^(6+6+6+6) = x^24 = x^(6*4)

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

To show the dfference between (x^6)*(x^4).

We know that x^6 = x.x.x.x.x.x.product of 6 x's

x^4 = x.x.x.x.  Product of 4x's

x^6*x^4 = (x.x.x.x.x.x)(x.x.x.x) = x^10. product of 10 x's.

(x^6)^4 = (x^6)(x^6)(x^6) (x^6) = (x.x.x.x.x.x)(x.x.x.x.x.x)(x.x.x.x.x.x.)(x.x.x.x.x.x) =  x^(6*4)  , product of 6 x's in 4 sets

= x^24 , product of 24 x's .

Hope this clears.

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