Sure, let's work through the problem step by step.
We'll start with the given expression:
[tex]\[ \frac{4m - 9n}{16m^2} - \frac{9n^2 + 1}{4m - 3n} \][/tex]
Step 1: Simplify each fraction if possible.
The first fraction is:
[tex]\[ \frac{4m - 9n}{16m^2} \][/tex]
This fraction cannot be simplified further in a straightforward manner, so we keep it as is for now.
The second fraction is:
[tex]\[ \frac{9n^2 + 1}{4m - 3n} \][/tex]
This fraction also appears to be in its simplest form in terms of factoring.
Step 2: Find a common denominator.
To subtract these two fractions, we need a common denominator. The common denominator will be the product of the two denominators:
[tex]\[ 16m^2 \cdot (4m - 3n) \][/tex]
Step 3: Rewrite each fraction with the common denominator.
For the first fraction [tex]\(\frac{4m - 9n}{16m^2}\)[/tex]:
[tex]\[ \frac{4m - 9n}{16m^2} \times \frac{4m - 3n}{4m - 3n} = \frac{(4m - 9n)(4m - 3n)}{16m^2(4m - 3n)} \][/tex]
For the second fraction [tex]\(\frac{9n^2 + 1}{4m - 3n}\)[/tex]:
[tex]\[ \frac{9n^2 + 1}{4m - 3n} \times \frac{16m^2}{16m^2} = \frac{(9n^2 + 1) \cdot 16m^2}{16m^2 \cdot (4m - 3n)} \][/tex]
Step 4: Perform the subtraction.
We can now subtract the numerators since the denominators are the same:
[tex]\[ \frac{(4m - 9n)(4m - 3n) - 16m^2(9n^2 + 1)}{16m^2 (4m - 3n)} \][/tex]
Let's expand the numerator:
[tex]\[ (4m - 9n)(4m - 3n) - 16m^2(9n^2 + 1) \][/tex]
First, expand [tex]\((4m - 9n)(4m - 3n)\)[/tex]:
[tex]\[ (4m)(4m) + (4m)(-3n) + (-9n)(4m) + (-9n)(-3n) \][/tex]
[tex]\[ = 16m^2 - 12mn - 36mn + 27n^2 \][/tex]
[tex]\[ = 16m^2 - 48mn + 27n^2 \][/tex]
Then, expand [tex]\(16m^2(9n^2 + 1)\)[/tex]:
[tex]\[ 16m^2 \cdot 9n^2 + 16m^2 \cdot 1 \][/tex]
[tex]\[ = 144m^2n^2 + 16m^2 \][/tex]
So the numerator becomes:
[tex]\[ 16m^2 - 48mn + 27n^2 - (144m^2n^2 + 16m^2) \][/tex]
Combine like terms:
[tex]\[ = 16m^2 - 48mn + 27n^2 - 144m^2n^2 - 16m^2 \][/tex]
[tex]\[ = 27n^2 - 48mn - 144m^2n^2 \][/tex]
Step 5: Simplify the expression if possible.
Now, our expression is:
[tex]\[ \frac{27n^2 - 48mn - 144m^2n^2}{16m^2(4m - 3n)} \][/tex]
Factor out the common terms in the numerator:
[tex]\[ = \frac{3n(-48mn - 16m + 9n)}{16m^2(4m - 3n)} \][/tex]
So, the final simplified form of the expression is:
[tex]\[ \frac{3n(-48m^2n - 16m + 9n)}{16m^2(4m - 3n)} \][/tex]
This is our answer.
