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Question Number 210605 by peter frank last updated on 13/Aug/24

Answered by A5T last updated on 13/Aug/24

a=2^x ;b=2^y ⇒a+b=18;(a/b)=5⇒a=5b ⇒6b=18⇒b=3⇒a=15 2^(x+y) =ab=15×3=45

$${a}=\mathrm{2}^{{x}} ;{b}=\mathrm{2}^{{y}} \Rightarrow{a}+{b}=\mathrm{18};\frac{{a}}{{b}}=\mathrm{5}\Rightarrow{a}=\mathrm{5}{b} \\ $$$$\Rightarrow\mathrm{6}{b}=\mathrm{18}\Rightarrow{b}=\mathrm{3}\Rightarrow{a}=\mathrm{15} \\ $$$$\mathrm{2}^{{x}+{y}} ={ab}=\mathrm{15}×\mathrm{3}=\mathrm{45} \\ $$

Commented by peter frank last updated on 14/Aug/24

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

Answered by klipto last updated on 15/Aug/24

k=2^x p=2^y ∴k+p=18 and from indices recall 2^x ×2^y =kp=2^(x+y) &2^x ÷2^y =(k/p)=2^(x−y) k+p=18 ,(k/p)=5 ,kp=? k=18−p then ((18−p)/p)=5,p=3 and k=18−3=15 ∴kp=15×3=45✓ klipto−quanta

$$ \\ $$$$ \\ $$$$\boldsymbol{\mathrm{k}}=\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:\:\:\boldsymbol{\mathrm{p}}=\mathrm{2}^{\boldsymbol{\mathrm{y}}} \\ $$$$\therefore\boldsymbol{\mathrm{k}}+\boldsymbol{\mathrm{p}}=\mathrm{18} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{from}}\:\boldsymbol{\mathrm{indices}}\:\boldsymbol{\mathrm{recall}} \\ $$$$\mathrm{2}^{\boldsymbol{\mathrm{x}}} ×\mathrm{2}^{\boldsymbol{\mathrm{y}}} =\boldsymbol{\mathrm{kp}}=\mathrm{2}^{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}} \:\&\mathrm{2}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\div}\mathrm{2}^{\boldsymbol{\mathrm{y}}} =\frac{\boldsymbol{\mathrm{k}}}{\boldsymbol{\mathrm{p}}}=\mathrm{2}^{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}} \\ $$$$\boldsymbol{\mathrm{k}}+\boldsymbol{\mathrm{p}}=\mathrm{18}\:\:,\frac{\boldsymbol{\mathrm{k}}}{\boldsymbol{\mathrm{p}}}=\mathrm{5}\:\:,\boldsymbol{\mathrm{kp}}=? \\ $$$$\boldsymbol{\mathrm{k}}=\mathrm{18}−\boldsymbol{\mathrm{p}}\:\: \\ $$$$\boldsymbol{\mathrm{then}} \\ $$$$\frac{\mathrm{18}−\boldsymbol{\mathrm{p}}}{\boldsymbol{\mathrm{p}}}=\mathrm{5},\boldsymbol{\mathrm{p}}=\mathrm{3}\: \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{k}}=\mathrm{18}−\mathrm{3}=\mathrm{15} \\ $$$$\therefore\boldsymbol{\mathrm{kp}}=\mathrm{15}×\mathrm{3}=\mathrm{45}\checkmark \\ $$$$\boldsymbol{\mathrm{klipto}}−\boldsymbol{\mathrm{quanta}} \\ $$

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