Certainly! We need to simplify the expression [tex]\((8x + 1)(8x - 1)\)[/tex]. Let's proceed step-by-step.
### Step-by-Step Solution
1. Identify the given expression:
We start with the product of two binomials: [tex]\((8x + 1)(8x - 1)\)[/tex].
2. Recognize the algebraic identity:
Notice that this expression is of the form [tex]\((a + b)(a - b)\)[/tex], which is a difference of squares. The difference of squares is a well-known algebraic identity given by:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
3. Assign variables to match the identity:
In our problem, we have:
[tex]\[
a = 8x \quad \text{and} \quad b = 1
\][/tex]
4. Apply the identity:
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the difference of squares formula:
[tex]\[
(8x + 1)(8x - 1) = (8x)^2 - 1^2
\][/tex]
5. Perform the squaring operations:
Calculate the square of each term:
[tex]\[
(8x)^2 = 64x^2 \quad \text{and} \quad 1^2 = 1
\][/tex]
6. Combine the results:
Putting it all together, we get:
[tex]\[
(8x + 1)(8x - 1) = 64x^2 - 1
\][/tex]
### Final Simplified Expression
Thus, the simplified form of the given expression [tex]\((8x + 1)(8x - 1)\)[/tex] is:
[tex]\[
64x^2 - 1
\][/tex]
This is the final answer.
