[tex]\frac{4n-m}{20mn^{2}} \frac{3m+n}{15^{2}n} \\\\\frac{b+3}{9b-27} - \frac{b-1}{b^{2}-3b}\\\\\frac{m^{2}-10mn+25n^{2}}{12m^{3}n^{2}} / \frac{m-5n}{6mn}\\\\\frac{a+3}{1-a} * (\frac{a}{a-3} + \frac{3-a}{a+3})[/tex]

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вопрос

To simplify each of the given expressions, we will break down the steps for each one:

(\frac{4n-m}{20mn^{2}} \times \frac{3m+n}{15^{2}n})

First, simplify each fraction individually:
(\frac{4n-m}{20mn^{2}} = \frac{n(4 - m)}{20mn^{2}} = \frac{4 - m}{20n})

(\frac{3m+n}{15^{2}n} = \frac{m + n}{225n} = \frac{m+n}{15^{2}n})

Now multiply the simplified fractions together:
(\frac{4 - m}{20n} \times \frac{m+n}{15^{2}n} = \frac{(4 - m)(m+n)}{20n \times 15^{2}n})

Simplify the expression further if possible.

(\frac{b+3}{9b-27} - \frac{b-1}{b^{2}-3b})

First, factor out common terms for each fraction:
(\frac{b+3}{9(b-3)} - \frac{b-1}{b(b-3)})

Now find a common denominator and combine the fractions:
(\frac{b(b+3) - 9(b-1)}{9b(b-3)})

Simplify the expression further by expanding and combining like terms.

(\frac{m^{2}-10mn+25n^{2}}{12m^{3}n^{2}} \div \frac{m-5n}{6mn})

Rewrite the division as multiplication by the reciprocal of the second fraction:
(\frac{m^{2}-10mn+25n^{2}}{12m^{3}n^{2}} \times \frac{6mn}{m-5n})

Expand and simplify the expression further.

(\frac{a+3}{1-a} \times \left(\frac{a}{a-3} + \frac{3-a}{a+3}\right))

Distribute the first fraction:
(\frac{a+3}{1-a} \times \frac{a}{a-3} + \frac{a+3}{1-a} \times \frac{3-a}{a+3})

Simplify each part of the expression and then combine the results.