64x^2 + 343

i know 8x * 8x will get you 64x^2, but i am confused on the rest of the problem.

(8x+sqrt (-343))(8x - sqrt(-343))

Try that.

To understand how to solve the expression 64x^2 + 343, let's break it down step-by-step.

First, it's important to note that the given expression is a sum of two terms: 64x^2 and 343. These terms cannot be directly combined or simplified together because they are not like terms. So, the expression is already in its simplest form.

However, if you are looking to factorize the expression, we can proceed further.

One approach is to recognize that 64x^2 can be expressed as the square of 8x, since (8x)^2 is equal to 64x^2. This is where your initial observation comes into play.

So, let's rewrite the expression using this information:

64x^2 + 343 = (8x)^2 + 343

Now, we need to focus on the second term, which is 343. This value can actually be written as the cube of 7:

343 = 7^3

With these two pieces of information, we can rewrite the expression as follows:

64x^2 + 343 = (8x)^2 + 7^3

Now, we have the form a^2 + b^3, which is a special algebraic identity called the sum of cubes. The sum of cubes identity states that a^3 + b^3 can be factorized as (a + b)(a^2 - ab + b^2).

Applying this identity to our expression, we get:

(8x)^2 + 7^3 = (8x + 7)((8x)^2 - (8x)(7) + 7^2)

Simplifying further:

(8x + 7)((8x)^2 - (56x) + 49)

So, the final factored form of the expression 64x^2 + 343 is:

(8x + 7)((8x)^2 - 56x + 49)

Note that the expression (8x)^2 - 56x + 49 cannot be factored further as it does not fit any common factorization patterns.