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Question-166215




Question Number 166215 by mathlove last updated on 15/Feb/22
Answered by nurtani last updated on 15/Feb/22
• 27^x −27^y  = 24 ⇔ 3^(3x) −3^(3y)  = 24 ⇔ (3^x )^3 −(3^y )^3 = 24  ⇔ (3^x −3^y )(3^(2x) +3^x ∙3^y +3^(2y) ) = 24  ⇔ (3^x −3^y )(3^(2x) +3^(x+y) +3^(2y) ) = 24  • 9^x +3^(x+y) +9^y  = 6 ⇔ 3^(2x) +3^(x+y) +3^(2y)  = 6  ⇒(3^x −3^y )(3^(2x) +3^(x+y) +3^(2y) ) = 24  (3^x −3^y )∙(6) = 24  3^x −3^y  = ((24)/6) = 4  ∴    3^x −3^y  = 4
$$\bullet\:\mathrm{27}^{{x}} −\mathrm{27}^{{y}} \:=\:\mathrm{24}\:\Leftrightarrow\:\mathrm{3}^{\mathrm{3}{x}} −\mathrm{3}^{\mathrm{3}{y}} \:=\:\mathrm{24}\:\Leftrightarrow\:\left(\mathrm{3}^{{x}} \right)^{\mathrm{3}} −\left(\mathrm{3}^{{y}} \right)^{\mathrm{3}} =\:\mathrm{24} \\ $$$$\Leftrightarrow\:\left(\mathrm{3}^{{x}} −\mathrm{3}^{{y}} \right)\left(\mathrm{3}^{\mathrm{2}{x}} +\mathrm{3}^{{x}} \centerdot\mathrm{3}^{{y}} +\mathrm{3}^{\mathrm{2}{y}} \right)\:=\:\mathrm{24} \\ $$$$\Leftrightarrow\:\left(\mathrm{3}^{{x}} −\mathrm{3}^{{y}} \right)\left(\mathrm{3}^{\mathrm{2}{x}} +\mathrm{3}^{{x}+{y}} +\mathrm{3}^{\mathrm{2}{y}} \right)\:=\:\mathrm{24} \\ $$$$\bullet\:\mathrm{9}^{{x}} +\mathrm{3}^{{x}+{y}} +\mathrm{9}^{{y}} \:=\:\mathrm{6}\:\Leftrightarrow\:\mathrm{3}^{\mathrm{2}{x}} +\mathrm{3}^{{x}+{y}} +\mathrm{3}^{\mathrm{2}{y}} \:=\:\mathrm{6} \\ $$$$\Rightarrow\left(\mathrm{3}^{{x}} −\mathrm{3}^{{y}} \right)\left(\mathrm{3}^{\mathrm{2}{x}} +\mathrm{3}^{{x}+{y}} +\mathrm{3}^{\mathrm{2}{y}} \right)\:=\:\mathrm{24} \\ $$$$\left(\mathrm{3}^{{x}} −\mathrm{3}^{{y}} \right)\centerdot\left(\mathrm{6}\right)\:=\:\mathrm{24} \\ $$$$\mathrm{3}^{{x}} −\mathrm{3}^{{y}} \:=\:\frac{\mathrm{24}}{\mathrm{6}}\:=\:\mathrm{4} \\ $$$$\therefore\:\:\:\:\mathrm{3}^{{x}} −\mathrm{3}^{{y}} \:=\:\mathrm{4} \\ $$$$\: \\ $$

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