Question

The expression (a3)2 can be simplified as shown here:

(a3)2 = (a3)(a3) = (a · a · a)(a · a · a) = a6

Write out in words how you would explain each step of this process to a friend. Why is each step equal to the previous step? Using a similar reasoning, explain why (a2)3= a6.

Answer 1

Answer and Explanation :

Given : The expression
`(a^3)^2` can be simplified as shown here:

`(a^3)^2 = (a^3)(a^3) = (a\cdot a \cdot a)(a \cdot a \cdot a) = a^6`

To find : Write out in words how you would explain each step of this process to a friend. Why is each step equal to the previous step?

Solution :

Step 1 - When we have a square term
`(a^3)^2` we have to multiply the term by itself.

So, We can write the term as
`(a^3)^2=(a^3)(a^3)`

Step 2 - When we have a cubic term
`(a^3)^2` we have to multiply the term two times by itself.

So, we can write the term as
`(a^3)(a^3)= (a\cdot a \cdot a)(a \cdot a \cdot a)`

Step 3 - When we have the same term repeat in multiple six times then the power of the term became 6.

So, we can write its as
`(a\cdot a \cdot a)(a \cdot a \cdot a)=a^6`

Therefore, following above steps we get,

`(a^3)^2 = (a^3)(a^3) = (a\cdot a \cdot a)(a \cdot a \cdot a) = a^6`

Answer 2

Given expression
`(a^3)^2`.

Let us simplify step by step with explaination.

Let us read the given expression.

`(a^3)^2` could be read as a power 3 and whole power 2.

Because we have whole power 2 of a^3. That represents two factors of (a^3) is there. So, we could write two factors of a^3 as (a^3)(a^3).

So, first step is (a^3) (a^3).

Now, each of the factor has a^3.

That is read as a power 3, that is three factors of a.

So, we can write a^3 in expanded form as a*a*a.

For each a^3, we would write a*a*a.

We have two facrors of a^3 there.

So, (a^3)(a^3) could be written as (a*a*a)(a*a*a).

There are 3+3=6 factors of a's there.

So, we could rewrite (a*a*a)(a*a*a) as 6 power of a.

That is a^6.

Therefore, we got the steps :

(a^3)^2 = (a^3)(a^3) = (a*a*a)(a*a*a) = a^6.