Certainly! Let's simplify the given expression step-by-step.
The expression we need to simplify is:
[tex]\[
\frac{9b^2 - 1}{b^2}
\][/tex]
Step 1: Recognize the structure of the numerator.
Notice that the numerator [tex]\(9b^2 - 1\)[/tex] is a difference of squares. Recall that the difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]. We can apply this formula to the numerator:
[tex]\[
9b^2 - 1 = (3b)^2 - 1^2
\][/tex]
This can be factored as:
[tex]\[
(3b - 1)(3b + 1)
\][/tex]
Step 2: Rewrite the entire expression with the factored form of the numerator.
Now, substitute the factored form back into the original expression:
[tex]\[
\frac{(3b - 1)(3b + 1)}{b^2}
\][/tex]
Step 3: Split the fraction.
We can separate the two terms in the numerator over the denominator [tex]\(b^2\)[/tex]:
[tex]\[
\frac{(3b - 1)(3b + 1)}{b^2} = \frac{3b - 1}{b} \cdot \frac{3b + 1}{b}
\][/tex]
This simplifies to:
[tex]\[
\left(\frac{3b - 1}{b}\right) \left(\frac{3b + 1}{b}\right)
\][/tex]
Step 4: Simplify each fraction separately.
Simplify each of the fractions inside the parentheses:
[tex]\[
\frac{3b - 1}{b} = 3 - \frac{1}{b} \quad \text{and} \quad \frac{3b + 1}{b} = 3 + \frac{1}{b}
\][/tex]
Step 5: Combine the simplified fractions.
Finally, multiply the simplified fractions together:
[tex]\[
(3 - \frac{1}{b})(3 + \frac{1}{b})
\][/tex]
So, the simplified form of the given expression [tex]\(\frac{9b^2 - 1}{b^2}\)[/tex] is:
[tex]\[
(3 - \frac{1}{b})(3 + \frac{1}{b})
\][/tex]
This is the fully simplified form of the given expression.
