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How can I evaluate $0.9^4$ without a calculator?

I think I have to use the binomial theorem but I don't know exactly how it works.

It should be in the form $(1-0.1)^4$.

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    $\begingroup$ I'd rather write $0.9=\frac 9{10}$. $\endgroup$
    – Git Gud
    Nov 21, 2014 at 0:53
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    $\begingroup$ Just a matter of terminology: one solves an equation, here you want to evaluate your expression. $\endgroup$
    – Lubin
    Nov 21, 2014 at 1:44
  • $\begingroup$ I can evaluate this in $8$ seconds in my head (definitely qualifies as not using a calculator) using the squaring number trick (which is easily Googleable). $\endgroup$
    – geometrian
    Nov 21, 2014 at 3:18
  • $\begingroup$ @Lubin Maybe he wants to solve the equation $x=(0.9)^4$! $\endgroup$
    – Joao
    Nov 21, 2014 at 3:59
  • $\begingroup$ @Joao That equation is already solved. The process of going from $(0.9)^4$ to a decimal number (or really any simpler form) is, in fact, called evaluating an expression :) $\endgroup$
    – Thomas
    Nov 21, 2014 at 5:18

4 Answers 4

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$$\begin{array}{llllll} &&&0.&9\\ \times&&&0.&9\\ \hline &&0.&8&1\\ \times&&&0.&9\\ \hline &0.&7&2&9\\ \times&&&0.&9\\ \hline 0.&6&5&6&1 \end{array}$$

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  • $\begingroup$ Where is the binomial method? I think you missed that while reading the question (if you 'actually' read it). $\endgroup$
    – Saharsh
    Nov 21, 2014 at 6:59
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    $\begingroup$ I did read the question; OP thought he/she has to use the binomial theorem, but I don't think using binomial theorem is the only way. $\endgroup$
    – peterwhy
    Nov 21, 2014 at 7:09
  • $\begingroup$ Of course there is no single way to solve the problem and that's the beauty of mathematics but do you think this answer have been a useful submission to the asker. Please don't consider me a revolutionist, it's just a comment after all. $\endgroup$
    – Saharsh
    Nov 21, 2014 at 7:14
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You said the trick yourself: $$0.9^4 = (1-0.1)^4 = 1^4 - 4\cdot 1^3 \cdot 0.1 + 6\cdot 1^2 \cdot 0.1^2 - 4 \cdot 1 \cdot 0.1^3 + 0.1^4,$$ which is easy to do. Another option is to notice that:

$$9^4 = 81^2 = (80 + 1)^2 = 6400 + 160 + 1 = 6561,$$ so we obtain: $$0.9^4 = (9 \cdot 10^{-1})^4 = 6561 \cdot 10^{-4} = 0.6561.$$

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    $\begingroup$ If someone can evaluate that without the use of a calculator, why cant they do so for 0.9 itself $\endgroup$ Nov 21, 2014 at 7:13
  • $\begingroup$ A lot of people have trouble with decimals.. $\endgroup$
    – Ivo Terek
    Nov 21, 2014 at 23:38
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$(0.9)^4=0.9\times 0.9\times 0.9\times 0.9=0.81\times 0.81=$

$\begin{array}{r}0.81\\\underline{\times 0.81}\\81\\\underline{6480}\\0.6561\end{array}$

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  • $\begingroup$ Simple multiplication is much easier than using the binomial theorem in this case. $\endgroup$
    – JRN
    Nov 21, 2014 at 1:04
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It may be not what you wanted, however, it is a good mathod or tool to solve such problem.

$$f(x_0+\Delta x) -f(x_0)\approx f'(x_0) \Delta x,$$

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  • $\begingroup$ Why negative vote? $\endgroup$
    – hola
    Nov 21, 2014 at 7:15

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