Reduce to the lowest terms:
Q16. $\dfrac{x^{2} - 1}{x^2 - x}$
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Numerator: $x^{2} - 1 = x^{2} - 1^{2} = (x + 1)(x - 1)$
Dinomerator: $x^2 - x = x(x - 1)$
$\therefore \dfrac{(x + 1)(x - 1)}{x(x - 1)} = \dfrac{x + 1}{x}$ Ans
Dinomerator: $x^2 - x = x(x - 1)$
$\therefore \dfrac{(x + 1)(x - 1)}{x(x - 1)} = \dfrac{x + 1}{x}$ Ans
Q17. $\dfrac{x - 2}{x^{2} + x - 6}$
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Numerator: $x - 2$
Dinomerator: $x^{2} + x - 6 = x^{2} + 3x - 2x - 6$
$\Rightarrow x^{2} + 3x - 2x - 6 = x(x + 3) - 2(x + 3)$
$\Rightarrow x(x + 3) - 2(x + 3) = (x + 3)(x - 2)$
$\therefore \dfrac{x - 2}{(x + 3)(x - 2)} = \dfrac{1}{x + 3}$ Ans
Dinomerator: $x^{2} + x - 6 = x^{2} + 3x - 2x - 6$
$\Rightarrow x^{2} + 3x - 2x - 6 = x(x + 3) - 2(x + 3)$
$\Rightarrow x(x + 3) - 2(x + 3) = (x + 3)(x - 2)$
$\therefore \dfrac{x - 2}{(x + 3)(x - 2)} = \dfrac{1}{x + 3}$ Ans
Q18. $\dfrac{3a^{3}b - 3ab^{2}}{a^{2} + ab}$
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Numerator: $3a^{3}b - 3ab^{2} = 3ab(a^{2} - b)$
Dinomerator: $a^{2} + ab = a(a + b)$
$\therefore \dfrac{3ab(a^{2} - b)}{a(a + b)} = \dfrac{3b(a^{2} - b)}{a + b}$ Ans
Dinomerator: $a^{2} + ab = a(a + b)$
$\therefore \dfrac{3ab(a^{2} - b)}{a(a + b)} = \dfrac{3b(a^{2} - b)}{a + b}$ Ans
Q19. $\dfrac{a^{2} - 9}{(a - 3)^{2}}$
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Numerator: $a^{2} - 3^{2} = (a + 3)(a - 3)$
Dinomerator: $(a - 3)^{2} = (a - 3)(a - 3)$
$\therefore \dfrac{(a + 3)(a - 3)}{(a - 3)(a - 3)} = \dfrac{a + 3}{a - 3}$ Ans
Dinomerator: $(a - 3)^{2} = (a - 3)(a - 3)$
$\therefore \dfrac{(a + 3)(a - 3)}{(a - 3)(a - 3)} = \dfrac{a + 3}{a - 3}$ Ans
Q20. $\dfrac{x^{2} - 3x}{x^{2} - 4x + 3}$
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Numerator: $x^{2} - 3x = x(x - 3)$
Dinomerator: $x^{2} - 4x + 3 = x^{2} - 3x - x + 3$
$\Rightarrow x^{2} - 3x - x + 3 = x(x - 3) - 1(x - 3)$
$\Rightarrow x(x - 3) - 1(x - 3) = (x - 3)(x - 1)$
$\therefore \dfrac{x(x - 3)}{(x - 3)(x - 1)} = \dfrac{x}{x - 1}$ Ans
Dinomerator: $x^{2} - 4x + 3 = x^{2} - 3x - x + 3$
$\Rightarrow x^{2} - 3x - x + 3 = x(x - 3) - 1(x - 3)$
$\Rightarrow x(x - 3) - 1(x - 3) = (x - 3)(x - 1)$
$\therefore \dfrac{x(x - 3)}{(x - 3)(x - 1)} = \dfrac{x}{x - 1}$ Ans
Q21. $\dfrac{a^{2} - a - 6}{a^{2} - 9}$
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Numerator: $a^{2} - a - 6 = a^{2} - 3a + 2a - 6$
$\Rightarrow a^{2} - 3a + 2a - 6 = a(a - 3) + 2(a - 3)$
$\Rightarrow a(a - 3) + 2(a - 3) = (a - 3)(a + 2)$
Dinomerator: $a^{2} - 9 = a^{2} - 3^{2} = (a + 3)(a - 3)$
$\therefore \dfrac{(a - 3)(a + 2)}{(a + 3)(a - 3)} = \dfrac{a + 2}{a + 3}$ Ans
$\Rightarrow a^{2} - 3a + 2a - 6 = a(a - 3) + 2(a - 3)$
$\Rightarrow a(a - 3) + 2(a - 3) = (a - 3)(a + 2)$
Dinomerator: $a^{2} - 9 = a^{2} - 3^{2} = (a + 3)(a - 3)$
$\therefore \dfrac{(a - 3)(a + 2)}{(a + 3)(a - 3)} = \dfrac{a + 2}{a + 3}$ Ans
Q22. $\dfrac{x^{2} - y^{2}}{2x^{2}y - 2xy^{2}}$
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Numerator: $x^{2} - y^{2} = (x + y)(x - y)$
Dinomerator: $2x^{2}y - 2xy^{2} = 2xy(x - y)$
$\therefore \dfrac{(x + y)(x - y)}{2xy(x - y)} = \dfrac{x + y}{2xy}$ Ans
Dinomerator: $2x^{2}y - 2xy^{2} = 2xy(x - y)$
$\therefore \dfrac{(x + y)(x - y)}{2xy(x - y)} = \dfrac{x + y}{2xy}$ Ans
Q23. $\dfrac{x^{2} - 4x - 21}{3x^{2} + 10x + 3}$
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Numerator: $x^{2} - 4x - 21 = x^{2} - 7x + 3x - 21$
$\Rightarrow x^{2} - 7x + 3x - 21 = x(x - 7) + 3(x - 7)$
$\Rightarrow x(x - 7) + 3x(x - 7) = (x - 7)(x + 3)$
Dinomerator: $3x^{2} + 10x + 3 = 3x^{2} + 9x + x + 3$
$\Rightarrow 3x^{2} + 9x + x + 3 = 3x(x + 3) + 1(x + 3)$
$\Rightarrow 3x(x + 3) + 1(x + 3) = (x + 3)(3x + 1)$
$\therefore \dfrac{(x - 7)(x + 3)}{(x + 3)(3x + 1)} = \dfrac{x - 7}{3x + 1}$ Ans
$\Rightarrow x^{2} - 7x + 3x - 21 = x(x - 7) + 3(x - 7)$
$\Rightarrow x(x - 7) + 3x(x - 7) = (x - 7)(x + 3)$
Dinomerator: $3x^{2} + 10x + 3 = 3x^{2} + 9x + x + 3$
$\Rightarrow 3x^{2} + 9x + x + 3 = 3x(x + 3) + 1(x + 3)$
$\Rightarrow 3x(x + 3) + 1(x + 3) = (x + 3)(3x + 1)$
$\therefore \dfrac{(x - 7)(x + 3)}{(x + 3)(3x + 1)} = \dfrac{x - 7}{3x + 1}$ Ans
Q24. $\dfrac{a^{3} - ab^{2}}{a^{3} + 2a^{2}b + ab^{2}}$
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Numerator: $a^{3} - ab^{2} = a(a^{2} - b^{2})$
$\Rightarrow a(a^{2} - b^{2}) = a(a + b)(a - b)$
Dinomerator: $a^{3} + 2a^{2}b + ab^{2} = a^{3} + a^{2}b + a^{2}b + ab^{2}$
$\Rightarrow a^{3} + a^{2}b + a^{2}b + ab^{2} = a^{2}(a + b) + ab(a + b)$
$\Rightarrow a^{2}(a + b) + ab(a + b) = (a + b)(a^{2} + ab)$
$\Rightarrow (a + b)(a^{2} + ab) = (a + b)a(a + b) = a(a + b)(a + b)$
$\therefore \dfrac{a(a + b)(a - b)}{a(a + b)(a + b)} = \dfrac{a - b}{a + b}$ Ans
$\Rightarrow a(a^{2} - b^{2}) = a(a + b)(a - b)$
Dinomerator: $a^{3} + 2a^{2}b + ab^{2} = a^{3} + a^{2}b + a^{2}b + ab^{2}$
$\Rightarrow a^{3} + a^{2}b + a^{2}b + ab^{2} = a^{2}(a + b) + ab(a + b)$
$\Rightarrow a^{2}(a + b) + ab(a + b) = (a + b)(a^{2} + ab)$
$\Rightarrow (a + b)(a^{2} + ab) = (a + b)a(a + b) = a(a + b)(a + b)$
$\therefore \dfrac{a(a + b)(a - b)}{a(a + b)(a + b)} = \dfrac{a - b}{a + b}$ Ans