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If $(1.001)^{1259} = 3.52$ and $(1.001)^{2062} = 7.85$, then $(1.001)^{3321}= ?$

what should be the approach in-order to get a solution without electronic aid?

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If you can't get electronic aid, then you can either use a mechanical computer or do the computation by hand. Another hint: $1259+2062=3321$ (done without electronic aid). And since in fact $1.001^{1259}\neq3.52$ (also checked by mere inspection), you can in actually derive any result you like (ex falso sequitur quodlibet: from a false hypothesis any conclusion can be drawn). –  Marc van Leeuwen Feb 12 '12 at 13:51
    
@MarcvanLeeuwen: How did you arrive at $1.001^{1259}\neq3.52$ by inspection? $1.001^{1259}\approx3.51968$, so $3.52$ isn't far off at all. –  Isaac Feb 12 '12 at 19:19
    
@Isaac: By the binomial formula, $1.001^{1259}$ has a digit $1$ at position $3777$ after the decimal point, while $3.52$ doesn't. So they differ. –  Marc van Leeuwen Feb 12 '12 at 20:16
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@MarcvanLeeuwen: Ahh, so the question would perhaps be better worded as "Since $(1.001)^{1259}\approx3.52$ and $(1.001)^{2062}\approx7.85$, $(1.001)^{3321}\approx ?$" –  Isaac Feb 12 '12 at 20:19
    
@Isaac: Most certainly. –  Marc van Leeuwen Feb 12 '12 at 20:20
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2 Answers

up vote 7 down vote accepted

$\begin{eqnarray} 1.001^{3321} &=& 1.001^{1259 + 2062} \\ &=& 1.001^{1259} \times 1.001^{2062}\end{eqnarray}$

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Cute, I don't know why haven't I thought of this. –  Quixotic Feb 12 '12 at 13:57
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$(1.001)^{3321}==(1.001)^{1259+2062}=>(1.001)^{1259}×(1.001)^{2062}$

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Welcome to Math.Stackexchange and thank you for wanting to participate. However, this question is more than a year old and has an answer already which is very similar to your answer. –  mrf Jun 9 '13 at 8:28
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