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Амалдарды орындандар: a) (m - 2n) : (2n - m) / 5mn; б) (a+5b)/(a2-5ab) : (25b2-a2)/5b2

Mathematics

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Here are the solutions to the problems.

a)

  1. Identify the denominators.

    The denominators are 4x−y4x-y and y−4xy-4x. Notice that y−4x=−(4x−y)y-4x = -(4x-y).

    3x+24x−y+2x−1y−4x=3x+24x−y+2x−1−(4x−y)\frac{3x+2}{4x-y} + \frac{2x-1}{y-4x} = \frac{3x+2}{4x-y} + \frac{2x-1}{-(4x-y)}
  2. Rewrite the expression with a common denominator.

    3x+24x−y−2x−14x−y\frac{3x+2}{4x-y} - \frac{2x-1}{4x-y}
  3. Combine the numerators over the common denominator.

    (3x+2)−(2x−1)4x−y=3x+2−2x+14x−y\frac{(3x+2) - (2x-1)}{4x-y} = \frac{3x+2-2x+1}{4x-y}
  4. Simplify the numerator.

    x+34x−y\frac{x+3}{4x-y}

    The final answer is x+34x−y\boxed{\frac{x+3}{4x-y}}.

    б)

  5. Factor the denominators in the first part of the expression.

    a2−5ab=a(a−5b)a^2-5ab = a(a-5b) a2+5ab=a(a+5b)a^2+5ab = a(a+5b)

    The expression becomes:

    (a+5ba(a−5b)−a−5ba(a+5b)):25b2−a25b2\left( \frac{a+5b}{a(a-5b)} - \frac{a-5b}{a(a+5b)} \right) : \frac{25b^2-a^2}{5b^2}
  6. Find a common denominator for the fractions inside the parenthesis, which is a(a−5b)(a+5b)a(a-5b)(a+5b).

    ((a+5b)(a+5b)a(a−5b)(a+5b)−(a−5b)(a−5b)a(a−5b)(a+5b)):25b2−a25b2\left( \frac{(a+5b)(a+5b)}{a(a-5b)(a+5b)} - \frac{(a-5b)(a-5b)}{a(a-5b)(a+5b)} \right) : \frac{25b^2-a^2}{5b^2} ((a+5b)2−(a−5b)2a(a−5b)(a+5b)):25b2−a25b2\left( \frac{(a+5b)^2 - (a-5b)^2}{a(a-5b)(a+5b)} \right) : \frac{25b^2-a^2}{5b^2}
  7. Expand the numerator using the difference of squares formula (X2−Y2)=(X−Y)(X+Y)(X^2-Y^2)=(X-Y)(X+Y), where X=a+5bX=a+5b and Y=a−5bY=a-5b.

    (a+5b)2−(a−5b)2=((a+5b)−(a−5b))((a+5b)+(a−5b))(a+5b)^2 - (a-5b)^2 = ((a+5b)-(a-5b))((a+5b)+(a-5b)) =(a+5b−a+5b)(a+5b+a−5b)=(10b)(2a)=20ab= (a+5b-a+5b)(a+5b+a-5b) = (10b)(2a) = 20ab

    The expression inside the parenthesis simplifies to:

    20aba(a−5b)(a+5b)=20b(a−5b)(a+5b)=20ba2−25b2\frac{20ab}{a(a-5b)(a+5b)} = \frac{20b}{(a-5b)(a+5b)} = \frac{20b}{a^2-25b^2}
  8. Perform the division.

    Division by a fraction is multiplication by its reciprocal. Also, note that 25b2−a2=−(a2−25b2)25b^2-a^2 = -(a^2-25b^2).

    20ba2−25b2:25b2−a25b2=20ba2−25b2×5b225b2−a2\frac{20b}{a^2-25b^2} : \frac{25b^2-a^2}{5b^2} = \frac{20b}{a^2-25b^2} \times \frac{5b^2}{25b^2-a^2} =20ba2−25b2×5b2−(a2−25b2)= \frac{20b}{a^2-25b^2} \times \frac{5b^2}{-(a^2-25b^2)}
  9. Multiply the fractions.

    20b⋅5b2−(a2−25b2)(a2−25b2)=100b3−(a2−25b2)2\frac{20b \cdot 5b^2}{-(a^2-25b^2)(a^2-25b^2)} = \frac{100b^3}{-(a^2-25b^2)^2} =−100b3(a2−25b2)2= -\frac{100b^3}{(a^2-25b^2)^2}

    The final answer is −100b3(a2−25b2)2\boxed{-\frac{100b^3}{(a^2-25b^2)^2}}.

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