Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 6302 by sanusihammed last updated on 22/Jun/16

Commented by Rasheed Soomro last updated on 22/Jun/16

3^x +4^x =25  Since 3^x , 4^x >0  ∀x∈Z  So,  3^x <25 ∧ 4^x <25 ⇒ x<3   Since 3^x +4^x  =25 (whole number)⇒x≥0       (If x were negative, 3^x , 4^x couldn′t be whole  and therefore their sum also.)  Now 0≤x<3  ∧ x∈Z  So possible values for x are only 0,1 & 2  0 and 1 do not satisfy the given equation  2 satisfies.  Hence x=2

$$\mathrm{3}^{{x}} +\mathrm{4}^{{x}} =\mathrm{25} \\ $$$$\mathrm{Since}\:\mathrm{3}^{\mathrm{x}} ,\:\mathrm{4}^{{x}} >\mathrm{0}\:\:\forall{x}\in\mathbb{Z} \\ $$$$\mathrm{So},\:\:\mathrm{3}^{\mathrm{x}} <\mathrm{25}\:\wedge\:\mathrm{4}^{{x}} <\mathrm{25}\:\Rightarrow\:{x}<\mathrm{3} \\ $$$$\:\mathrm{Since}\:\mathrm{3}^{\mathrm{x}} +\mathrm{4}^{\mathrm{x}} \:=\mathrm{25}\:\left(\mathrm{whole}\:\mathrm{number}\right)\Rightarrow{x}\geqslant\mathrm{0}\:\:\:\:\: \\ $$$$\left(\mathrm{If}\:\mathrm{x}\:\mathrm{were}\:\mathrm{negative},\:\mathrm{3}^{{x}} ,\:\mathrm{4}^{{x}} \mathrm{couldn}'\mathrm{t}\:\mathrm{be}\:\mathrm{whole}\right. \\ $$$$\left.\mathrm{and}\:\mathrm{therefore}\:\mathrm{their}\:\mathrm{sum}\:\mathrm{also}.\right) \\ $$$$\mathrm{Now}\:\mathrm{0}\leqslant\mathrm{x}<\mathrm{3}\:\:\wedge\:\mathrm{x}\in\mathbb{Z} \\ $$$$\mathrm{So}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{for}\:\mathrm{x}\:\mathrm{are}\:\mathrm{only}\:\mathrm{0},\mathrm{1}\:\&\:\mathrm{2} \\ $$$$\mathrm{0}\:\mathrm{and}\:\mathrm{1}\:\mathrm{do}\:\mathrm{not}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{given}\:\mathrm{equation} \\ $$$$\mathrm{2}\:\mathrm{satisfies}. \\ $$$$\mathrm{Hence}\:\mathrm{x}=\mathrm{2} \\ $$

Commented by Yozzii last updated on 22/Jun/16

x=2, unsure of working

$${x}=\mathrm{2},\:{unsure}\:{of}\:{working} \\ $$

Commented by FilupSmith last updated on 23/Jun/16

This isn′t really a good answer,  but if you recognise it as:  3^x +4^x =5^2   You can recognise it as a “3−4−5”  right angle triangle making x=2

$$\mathrm{This}\:\mathrm{isn}'\mathrm{t}\:\mathrm{really}\:\mathrm{a}\:\mathrm{good}\:\mathrm{answer}, \\ $$$$\mathrm{but}\:\mathrm{if}\:\mathrm{you}\:\mathrm{recognise}\:\mathrm{it}\:\mathrm{as}: \\ $$$$\mathrm{3}^{{x}} +\mathrm{4}^{{x}} =\mathrm{5}^{\mathrm{2}} \\ $$$$\mathrm{You}\:\mathrm{can}\:\mathrm{recognise}\:\mathrm{it}\:\mathrm{as}\:\mathrm{a}\:``\mathrm{3}−\mathrm{4}−\mathrm{5}'' \\ $$$$\mathrm{right}\:\mathrm{angle}\:\mathrm{triangle}\:\mathrm{making}\:{x}=\mathrm{2} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com