[tex]\bf \cfrac{12(x+y)^3}{9(x+y)}\implies \cfrac{12}{9}\cdot \cfrac{(x+y)(x+y)(x+y)}{(x+y)}
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\cfrac{3\cdot 4}{3\cdot 3}\cdot \cfrac{(x+y)(x+y)(x+y)}{(x+y)}\implies \cfrac{4}{3}\cfrac{(x+y)(x+y)}{1}\implies \cfrac{4(x+y)^2}{3}\\\\
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\cfrac{3x\sqrt{x}}{x^{\frac{1}{2}}}\impliedby recall\to a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}}\qquad thus\quad \cfrac{3x\sqrt{x}}{\sqrt{x}}[/tex]
[tex]\bf 3x\cdot \cfrac{\sqrt{x}}{\sqrt{x}}\implies 3x\cdot 1\implies 3x\\\\
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\left( \cfrac{\sqrt{2}\sqrt{x^3}}{\sqrt{x}} \right)^4\impliedby \textit{now, let's distribute the exponent}[/tex]
[tex]\bf \left( \cfrac{(\sqrt{2})^4(\sqrt{x^3})^4}{(\sqrt{x})^4} \right)\implies \cfrac{\sqrt{2^4}\sqrt{x^{3\cdot 4}}}{\sqrt{x^4}}\implies \cfrac{\sqrt{(2^2)^2}\sqrt{x^{12}}}{\sqrt{(x^2)^2}}\implies \cfrac{2^2\sqrt{x^{6\cdot 2}}}{x^2}
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\cfrac{2^2\sqrt{(x^6)^2}}{x^2}\implies \cfrac{2^2x^6}{x^2}\implies 4x^6x^{-2}\implies 4x^{6-2}\implies 4x^4[/tex]
