To solve the expression [tex]\(\frac{4m - 9n}{16m^2} - \frac{9n^2 + 1}{4m - 3n}\)[/tex], we need to follow several steps to simplify it. Let's break down the problem step-by-step.
Step 1: Identifying and Rewriting the Expressions
We have two fractions:
1. [tex]\(\frac{4m - 9n}{16m^2}\)[/tex]
2. [tex]\(\frac{9n^2 + 1}{4m - 3n}\)[/tex]
To subtract these fractions, we'll need a common denominator.
Step 2: Finding a Common Denominator
The denominators in the two fractions are [tex]\(16m^2\)[/tex] and [tex]\(4m - 3n\)[/tex]. To find a common denominator, we can multiply these two expressions together:
[tex]\[
16m^2 \cdot (4m - 3n)
\][/tex]
Step 3: Rewriting Each Fraction with the Common Denominator
Each term must be rewritten to have the common denominator.
For the first fraction:
[tex]\[
\frac{4m - 9n}{16m^2} \cdot \frac{4m - 3n}{4m - 3n} = \frac{(4m - 9n)(4m - 3n)}{16m^2(4m - 3n)}
\][/tex]
For the second fraction:
[tex]\[
\frac{9n^2 + 1}{4m - 3n} \cdot \frac{16m^2}{16m^2} = \frac{(9n^2 + 1) \cdot 16m^2}{16m^2(4m - 3n)}
\][/tex]
Step 4: Performing the Subtraction
Now we have:
[tex]\[
\frac{(4m - 9n)(4m - 3n)}{16m^2(4m - 3n)} - \frac{16m^2(9n^2 + 1)}{16m^2(4m - 3n)}
\][/tex]
Both fractions share the same denominator, so we can subtract the numerators directly:
[tex]\[
\frac{(4m - 9n)(4m - 3n) - 16m^2(9n^2 + 1)}{16m^2(4m - 3n)}
\][/tex]
Step 5: Simplifying the Numerator
Let's expand and simplify the numerator.
Expanding [tex]\((4m - 9n)(4m - 3n)\)[/tex]:
[tex]\[
(4m - 9n)(4m - 3n) = 16m^2 - 12mn - 36mn + 27n^2 = 16m^2 - 48mn + 27n^2
\][/tex]
So the numerator becomes:
[tex]\[
16m^2 - 48mn + 27n^2 - 16m^2(9n^2 + 1)
\][/tex]
Distribute [tex]\(16m^2\)[/tex] in the second term:
[tex]\[
16m^2 - 48mn + 27n^2 - 144m^2n^2 - 16m^2
\][/tex]
Combine like terms:
[tex]\[
-48mn + 27n^2 - 144m^2n^2
\][/tex]
We can factor out a common factor from the above expression, specifically [tex]\(3n\)[/tex]:
[tex]\[
3n(-48mn - 16m + 9n)
\][/tex]
Step 6: Writing the Final Expression
Putting it all together, our final expression is:
[tex]\[
\frac{3n(-48mn - 16m + 9n)}{16m^2(4m - 3n)}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{3n(-48mn - 16m + 9n)}{16m^2(4m - 3n)}
\][/tex]
