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Simplify the expression:

[tex]\[
\frac{4m - 9n}{16m^2} - \frac{9n^2 + 1}{4m - 3n}
\][/tex]


Sagot :

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]
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