Question

complex fractions
a complex fraction contains \smaller\fraction/s in its numerator or denominator, which are separated by the longest \fraction bar\ (vinculum) in the expression. simplifying a complex fraction involves simplifying its numerator and denominator and then performing division between them.
\\(\frac{\frac{5x^2}{2}}{\frac{15x^7}{8}}\\) \\(\frac{\frac{3}{b^8}}{\frac{6}{b^7}}\\) \\(\frac{\frac{r - 2}{2r^3}}{\frac{2 - r}{10r^2}}\\)
\\(\frac{n^3}{\frac{3n - 1}{\frac{n}{1 - 3n}}}\\) \\(\frac{\frac{y^3}{y^2 - 9}}{\frac{y^2}{y + 3}}\\) \\(\frac{\frac{a - 1}{a}}{\frac{a^2 - 1}{a^3}}\\)
\\(\frac{7 - \frac{5}{p - 3}}{\frac{7}{p - 3} + 5}\\) \\(\frac{\frac{2}{m + 1}}{2 + \frac{3}{m + 1}}\\) \\(\frac{\frac{z^2 - 25}{3}}{z^2 + 2z - 15}\\)
\\(\frac{\frac{u^2 + 5u + 6}{5u + 15}}{u}\\) \\(\frac{\frac{3}{x - 2} - \frac{2}{x + 1}}{\frac{x + 7}{x - 2}}\\) \\(\frac{\frac{1}{n} - \frac{n}{2}}{\frac{n}{2} - \frac{1}{n}}\\)

Explanation:

Step1: Simplify $\frac{\frac{5x^2}{2}}{\frac{15x^7}{8}}$

Rewrite as multiplication by reciprocal:
$\frac{5x^2}{2} \times \frac{8}{15x^7} = \frac{5 \times 8 \times x^2}{2 \times 15 \times x^7} = \frac{40}{30}x^{2-7} = \frac{4}{3x^5}$

Step2: Simplify $\frac{\frac{3}{b^5}}{\frac{6}{b^7}}$

Rewrite as multiplication by reciprocal:
$\frac{3}{b^5} \times \frac{b^7}{6} = \frac{3b^7}{6b^5} = \frac{1}{2}b^{7-5} = \frac{b^2}{2}$

Step3: Simplify $\frac{\frac{x-2}{2r^5}}{\frac{2-x}{10r^2}}$

Rewrite as multiplication by reciprocal, note $2-x=-(x-2)$:
$\frac{x-2}{2r^5} \times \frac{10r^2}{-(x-2)} = \frac{10r^2}{-2r^5} = -5r^{2-5} = -\frac{5}{r^3}$

Step4: Simplify $\frac{\frac{n^3}{3n-1}}{\frac{n}{1-3n}}$

Rewrite as multiplication by reciprocal, note $1-3n=-(3n-1)$:
$\frac{n^3}{3n-1} \times \frac{-(3n-1)}{n} = -n^{3-1} = -n^2$

Step5: Simplify $\frac{\frac{y^3}{y^2-9}}{\frac{y^2}{y+3}}$

Factor $y^2-9=(y+3)(y-3)$, rewrite as multiplication:
$\frac{y^3}{(y+3)(y-3)} \times \frac{y+3}{y^2} = \frac{y^3(y+3)}{y^2(y+3)(y-3)} = \frac{y}{y-3}$

Step6: Simplify $\frac{\frac{a-1}{a}}{\frac{a^2-1}{a^2}}$

Factor $a^2-1=(a+1)(a-1)$, rewrite as multiplication:
$\frac{a-1}{a} \times \frac{a^2}{(a+1)(a-1)} = \frac{a^2(a-1)}{a(a+1)(a-1)} = \frac{a}{a+1}$

Step7: Simplify $\frac{7-\frac{5}{p-3}}{\frac{7}{p-3}+5}$

Multiply numerator/denominator by $p-3$:
$\frac{(7)(p-3)-5}{7+5(p-3)} = \frac{7p-21-5}{7+5p-15} = \frac{7p-26}{5p-8}$

Step8: Simplify $\frac{\frac{2}{m+1}}{2+\frac{3}{m+1}}$

Multiply numerator/denominator by $m+1$:
$\frac{2}{2(m+1)+3} = \frac{2}{2m+2+3} = \frac{2}{2m+5}$

Step9: Simplify $\frac{\frac{z^2-25}{3}}{z^2+2x-15}$

Factor $z^2-25=(z+5)(z-5)$, rewrite as division:
$\frac{(z+5)(z-5)}{3(z^2+2z-15)} = \frac{(z+5)(z-5)}{3(z+5)(z-3)} = \frac{z-5}{3(z-3)}$

Step10: Simplify $\frac{\frac{u^2+5u+6}{5u+15}}{u}$

Factor numerator and denominator:
$\frac{(u+2)(u+3)}{5(u+3)} \times \frac{1}{u} = \frac{u+2}{5u}$

Step11: Simplify $\frac{\frac{3}{x-2}-\frac{2}{x+1}}{\frac{x+7}{x-2}}$

Combine numerator fractions, rewrite as multiplication:
First, numerator: $\frac{3(x+1)-2(x-2)}{(x-2)(x+1)} = \frac{3x+3-2x+4}{(x-2)(x+1)} = \frac{x+7}{(x-2)(x+1)}$
Multiply by reciprocal of denominator:
$\frac{x+7}{(x-2)(x+1)} \times \frac{x-2}{x+7} = \frac{1}{x+1}$

Step12: Simplify $\frac{\frac{1}{n}-n}{\frac{n}{2}-\frac{1}{n}}$

Multiply numerator/denominator by $2n$ to eliminate fractions:
$\frac{2 - 2n^2}{n^2 - 2} = \frac{-2(n^2-1)}{n^2-2} = -\frac{2(n^2-1)}{n^2-2}$

Answer:

1. $\frac{4}{3x^5}$
2. $\frac{b^2}{2}$
3. $-\frac{5}{r^3}$
4. $-n^2$
5. $\frac{y}{y-3}$
6. $\frac{a}{a+1}$
7. $\frac{7p-26}{5p-8}$
8. $\frac{2}{2m+5}$
9. $\frac{z-5}{3(z-3)}$
10. $\frac{u+2}{5u}$
11. $\frac{1}{x+1}$
12. $-\frac{2(n^2-1)}{n^2-2}$