Question

name neysindee velez

1. \\(\dfrac{\dfrac{u}{2} - \dfrac{u}{u - 4}}{\dfrac{4}{u - 4} - \dfrac{25}{2}}\\)
2. \\(\dfrac{\dfrac{1}{5} + \dfrac{1}{3}}{\dfrac{25}{3} + \dfrac{a^2}{a + 1}}\\)
3. \\(\dfrac{\dfrac{x}{9} + \dfrac{4}{5}}{\dfrac{2}{3} + \dfrac{x}{5}}\\)
4. \\(\dfrac{\dfrac{9}{m} - \dfrac{3}{m + 3}}{\dfrac{m - 1}{3} + \dfrac{m}{m + 3}}\\)

Answer:

Let's solve problem 17 step by step. The problem is a complex fraction: \(\frac{\frac{u}{2}-\frac{u}{u - 4}}{\frac{4}{u - 4}-\frac{25}{2}}\)

Step 1: Simplify the numerator \(\frac{u}{2}-\frac{u}{u - 4}\)

To subtract these two fractions, we need a common denominator, which is \(2(u - 4)\).

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Step 2: Simplify the denominator \(\frac{4}{u - 4}-\frac{25}{2}\)

The common denominator here is \(2(u - 4)\).

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Step 3: Divide the simplified numerator by the simplified denominator

Dividing by a fraction is the same as multiplying by its reciprocal. So we have:

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The \(2(u - 4)\) terms cancel out:

$$ \frac{u(u - 6)}{108-25u} $$

Final Answer for Problem 17:

\(\boxed{\frac{u(u - 6)}{108 - 25u}}\)

Now let's solve problem 18: \(\frac{\frac{1}{5}+\frac{1}{3}}{\frac{25}{3}+\frac{a^{2}}{a + 1}}\)

Step 1: Simplify the numerator \(\frac{1}{5}+\frac{1}{3}\)

The common denominator is \(15\).

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Step 2: Simplify the denominator \(\frac{25}{3}+\frac{a^{2}}{a + 1}\)

The common denominator is \(3(a + 1)\).

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Step 3: Divide the simplified numerator by the simplified denominator

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Final Answer for Problem 18:

\(\boxed{\frac{8(a + 1)}{5(3a^{2}+25a + 25)}}\)

Let's solve problem 19: \(\frac{\frac{x}{9}+\frac{4}{5}}{\frac{2}{3}+\frac{x}{5}}\)

Step 1: Simplify the numerator \(\frac{x}{9}+\frac{4}{5}\)

The common denominator is \(45\).

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Step 2: Simplify the denominator \(\frac{2}{3}+\frac{x}{5}\)

The common denominator is \(15\).

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Step 3: Divide the simplified numerator by the simplified denominator

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Final Answer for Problem 19:

\(\boxed{\frac{5x + 36}{3(3x + 10)}}\)

Let's solve problem 20: \(\frac{\frac{9}{m}-\frac{3}{m + 3}}{\frac{m - 1}{3}+\frac{m}{m + 3}}\)

Step 1: Simplify the numerator \(\frac{9}{m}-\frac{3}{m + 3}\)

The common denominator is \(m(m + 3)\).

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Step 2: Simplify the denominator \(\frac{m - 1}{3}+\frac{m}{m + 3}\)

The common denominator is \(3(m + 3)\).

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Step 3: Divide the simplified numerator by the simplified denominator

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Final Answer for Problem 20:

\(\boxed{\frac{9(2m + 9)}{m(m^{2}+5m - 3)}}\)