Question

Determine whether 49x^6 − 81b^2 is a difference of two squares. If so, factor it. If not, explain why.

Answer 1

Yes;
`(7x^3+9b)(7x^3-9b)`

Explanation:

A difference of squares is basically where you have two terms that can both be written in the form of
`k^2` and one is subtracting the other.

Here, the first term is
`49x^6` . We see that if we square root this, we will come out with the clean number:
`7x^3` (because
`7x^3*7x^3=49x^6`).

The second term is
`81b^2` . Again, we see that if we square root this, we will get the clean result:
`9b` (because
`9b*9b=81b^2`).

So, they are indeed both squares; thus, this is a difference of squares.

To factor it, we remember the formula for factoring
`a^2-b^2`:
`a^2-b^2=(a+b)(a-b)`

In this case, a = 7x^3 and b = 9b, so:

`49x^6-81b^2=(7x^3+9b)(7x^3-9b)`, and that's the answer.

Hope that helps!