show (prove) in detail why (2.3.4)^4=2^4 3^4 4^4. This is numbers in exponential form raised to a power

Expert Answers
ishpiro eNotes educator| Certified Educator

You have to prove that an exponent of a product (4th power of `2*3*4` ) equals the product of the 4th powers of the individual factors:

`(2*3*4)^4 = 2^4 * 3^4 * 4^4`

To prove this, let's apply the definition of exponent to the expression on the left side, `(2*3*4)^4` .

By definition, to take exponent, or raise the number or expression to a given power, means to multiply that number or expression by itself the number of times indicated by exponent. So, to raise `(2*3*4)` to the 4th power, we need to multiply this product by itself 4 times:

`(2*3*4)^4 = (2*3*4)*(2*3*4)*(2*3*4)*(2*3*4)`

Commutative and associative properties of multiplication state that one can multiply numbers in any order. So, we can rearrange the factors above in any way, and we can rearrange it so 2s are next to each other, 3s are next to each other, and 4s are next to each other:

`(2*3*4)^4 = (2*2*2*2)*(3*3*3*3)*(4*4*4*4)`

In the first parenthesis on the right side, according the definition of exponent, again, if 2 is multiplied by itself 4 times, it equals 2 to the 4th power, or `2^4` . Similarly, the second parenthesis contains `3^4` and the third parenthesis contains `4^4` .

Therefore, `(2*3*4)^4 = 2^4 * 3^4* 4^4` .

Hope this helps.

mrsfaragher | Student

What you are being asked to prove is that you can "distribute" exponents over multiplication.  This a a true property of exponents.  I have written this so it is easy to see how to write it as a 2-column proof, but I did add explanations between each step.

Prove: ` (2*3*4)^4=2^4*3^4*4^4`

First, lets expand the left using the definition of exponents (Ex: `x^3=x*x*x` )``

1.  `(2*3*4)^4=(2*3*4)(2*3*4)(2*3*4)(2*3*4)`

Reason: definition of exponents

Next, lets take the answer from the first step and group the numbers that are similar together.  We can do this because all the numbers are being multiplied together, and order doesn't matter in multiplication (its called the commutative property).  Parenthesis just mean multiplication in this case.

2.  `(2*3*4)(2*3*4)(2*3*4)(2*3*4)=2*2*2*2*3*3*3*3*4*4*4*4`

Reason:  commutative property of multiplication

Just to make things easier to read, lets parenthesis around the numbers that are the same.  We can do this because of the associative property of multiplication.  Again, the parenthesis are just saying multiply, but it makes it easier to read.

3.  `2*2*2*2*3*3*3*3*4*4*4*4=(2*2*2*2)*(3*3*3*3)*(4*4*4*4)`

Reason:  associative property of multiplication

Lastly, we can re-write what we have in exponent form because of the definition of exponents.

4.   `(2*2*2*2)*(3*3*3*3)*(4*4*4*4)=2^4*3^4*4^4`

Reason:  definition of exponents

It should be fairly obvious to most people at this point that we have proved what we set out to, but since this is a proof, we need to state what we were originally trying to prove again.  Proofs are just detailed explanations about why things in math work.  We can make the final statement because of a property called transitivity (if a=b, and b=c, then a=c).

5.  `(2*3*4)^4=2^4*3^4*4^4`

Reason:  transitivity

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