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Question Number 53935 by Mikael_Marshall last updated on 27/Jan/19

(√8^(x−2) )×(4^(2x−3) )^(1/(x+1)) =(2^(5x+5) )^(1/6)

$$\sqrt{\mathrm{8}^{{x}−\mathrm{2}} }×\sqrt[{{x}+\mathrm{1}}]{\mathrm{4}^{\mathrm{2}{x}−\mathrm{3}} }=\sqrt[{\mathrm{6}}]{\mathrm{2}^{\mathrm{5}{x}+\mathrm{5}} } \\ $$$$ \\ $$

Answered by Kunal12588 last updated on 27/Jan/19

2^((3x−6)/2) ×2^((4x−6)/(x+1)) =2^((5x+5)/6) ⇒((3x−6)/2)+((4x−6)/(x+1))=((5x+5)/6) ⇒((4x−6)/(x+1))=((5x+5)/6)−((9x−18)/2)=((5x+5−9x+18)/2) ⇒((4x−6)/(x+1))=((23−4x)/6) ⇒24x−36=23x−4x^2 +23−4x ⇒4x^2 +5x−59=0 comparing with std. form of quad. eq^n . a=4, b=5, c=−59 x=((−b±(√(b^2 −4ac)))/(2a))=((−5±(√(25+944)))/8) x=((±(√(969))−5)/8)=((±(√3)(√(17))(√(19))−5)/8) (use calculator)

$$\mathrm{2}^{\frac{\mathrm{3}{x}−\mathrm{6}}{\mathrm{2}}} ×\mathrm{2}^{\frac{\mathrm{4}{x}−\mathrm{6}}{{x}+\mathrm{1}}} =\mathrm{2}^{\frac{\mathrm{5}{x}+\mathrm{5}}{\mathrm{6}}} \\ $$$$\Rightarrow\frac{\mathrm{3}{x}−\mathrm{6}}{\mathrm{2}}+\frac{\mathrm{4}{x}−\mathrm{6}}{{x}+\mathrm{1}}=\frac{\mathrm{5}{x}+\mathrm{5}}{\mathrm{6}} \\ $$$$\Rightarrow\frac{\mathrm{4}{x}−\mathrm{6}}{{x}+\mathrm{1}}=\frac{\mathrm{5}{x}+\mathrm{5}}{\mathrm{6}}−\frac{\mathrm{9}{x}−\mathrm{18}}{\mathrm{2}}=\frac{\mathrm{5}{x}+\mathrm{5}−\mathrm{9}{x}+\mathrm{18}}{\mathrm{2}} \\ $$$$\Rightarrow\frac{\mathrm{4}{x}−\mathrm{6}}{{x}+\mathrm{1}}=\frac{\mathrm{23}−\mathrm{4}{x}}{\mathrm{6}} \\ $$$$\Rightarrow\mathrm{24}{x}−\mathrm{36}=\mathrm{23}{x}−\mathrm{4}{x}^{\mathrm{2}} +\mathrm{23}−\mathrm{4}{x} \\ $$$$\Rightarrow\mathrm{4}{x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{59}=\mathrm{0} \\ $$$${comparing}\:{with}\:{std}.\:{form}\:{of}\:{quad}.\:{eq}^{{n}} . \\ $$$${a}=\mathrm{4},\:{b}=\mathrm{5},\:{c}=−\mathrm{59} \\ $$$${x}=\frac{−{b}\pm\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}}}{\mathrm{2}{a}}=\frac{−\mathrm{5}\pm\sqrt{\mathrm{25}+\mathrm{944}}}{\mathrm{8}} \\ $$$${x}=\frac{\pm\sqrt{\mathrm{969}}−\mathrm{5}}{\mathrm{8}}=\frac{\pm\sqrt{\mathrm{3}}\sqrt{\mathrm{17}}\sqrt{\mathrm{19}}−\mathrm{5}}{\mathrm{8}}\:\:\:\left({use}\:{calculator}\right) \\ $$

Commented by Kunal12588 last updated on 27/Jan/19

3.2660956 and −4.5160956 from calculator

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