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Question Number 199900 by Calculusboy last updated on 11/Nov/23

Answered by Rasheed.Sindhi last updated on 11/Nov/23

((((3^x +7^(−x) ))^(1/x) +((3^(−x) +7^x ))^(1/x) )/( ((21^x +1))^(1/x) )) =(((((21^x +1)/7^x ))^(1/x) +(((1+21^x )/3^x ))^(1/x) )/( ((21^x +1))^(1/x) )) =(((((21^x +1))^(1/x) /7) +(((21^x +1))^(1/x) /3) )/( ((21^x +1))^(1/x) )) =(( ((3((21^x +1))^(1/x) +7((21^x +1))^(1/x) )/(21)) )/( ((21^x +1))^(1/x) )) =((10)/(21))

$$\frac{\sqrt[{{x}}]{\mathrm{3}^{{x}} +\mathrm{7}^{−{x}} }\:+\sqrt[{{x}}]{\mathrm{3}^{−{x}} +\mathrm{7}^{{x}} }\:}{\:\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}} \\ $$$$=\frac{\sqrt[{{x}}]{\frac{\mathrm{21}^{{x}} +\mathrm{1}}{\mathrm{7}^{{x}} }}\:+\sqrt[{{x}}]{\frac{\mathrm{1}+\mathrm{21}^{{x}} }{\mathrm{3}^{{x}} }}\:}{\:\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}} \\ $$$$=\frac{\frac{\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}}{\mathrm{7}}\:+\frac{\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}}{\mathrm{3}}\:}{\:\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}} \\ $$$$=\frac{\:\frac{\mathrm{3}\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}\:+\mathrm{7}\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}}{\mathrm{21}}\:\:}{\:\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}} \\ $$$$=\frac{\mathrm{10}}{\mathrm{21}} \\ $$

Answered by Rasheed.Sindhi last updated on 11/Nov/23

((((3^x +7^(−x) ))^(1/x) +((3^(−x) +7^x ))^(1/x) )/( ((21^x +1))^(1/x) )) =(((3^x (1+(1/(3^x .7^x ))))/(21^x (1+(1/(21^x ))))))^(1/x) +(((7^x ((1/(3^x .7^x ))+1))/(21^x (1+(1/(21^x ))))))^(1/x) =(((3^x (1+(1/(21^x ))))/(21^x (1+(1/(21^x ))))))^(1/x) +(((7^x (1+(1/(21^x ))))/(21^x (1+(1/(21^x ))))))^(1/x) =((3+7)/(21)) =((10)/(21))

$$\frac{\sqrt[{{x}}]{\mathrm{3}^{{x}} +\mathrm{7}^{−{x}} }\:+\sqrt[{{x}}]{\mathrm{3}^{−{x}} +\mathrm{7}^{{x}} }\:}{\:\sqrt[{{x}}]{\mathrm{21}^{{x}} +\mathrm{1}}\:} \\ $$$$=\sqrt[{{x}}]{\frac{\mathrm{3}^{{x}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}^{{x}} .\mathrm{7}^{{x}} }\right)}{\mathrm{21}^{{x}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{21}^{{x}} }\right)}}\:+\sqrt[{{x}}]{\frac{\mathrm{7}^{{x}} \left(\frac{\mathrm{1}}{\mathrm{3}^{{x}} .\mathrm{7}^{{x}} }+\mathrm{1}\right)}{\mathrm{21}^{{x}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{21}^{{x}} }\right)}} \\ $$$$=\sqrt[{{x}}]{\frac{\mathrm{3}^{{x}} \cancel{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{21}^{{x}} }\right)}}{\mathrm{21}^{{x}} \cancel{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{21}^{{x}} }\right)}}}\:+\sqrt[{{x}}]{\frac{\mathrm{7}^{{x}} \cancel{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{21}^{{x}} }\right)}}{\mathrm{21}^{{x}} \cancel{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{21}^{{x}} }\right)}}} \\ $$$$=\frac{\mathrm{3}+\mathrm{7}}{\mathrm{21}} \\ $$$$=\frac{\mathrm{10}}{\mathrm{21}} \\ $$

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