# How to evaluate $(0.9)^4$ without calculator

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 $(1-0.1)^4$.

• I'd rather write $0.9=\frac 9{10}$. Nov 21, 2014 at 0:53
• Just a matter of terminology: one solves an equation, here you want to evaluate your expression. Nov 21, 2014 at 1:44
• 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). Nov 21, 2014 at 3:18
• @Lubin Maybe he wants to solve the equation $x=(0.9)^4$!
– Joao
Nov 21, 2014 at 3:59
• @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 :) Nov 21, 2014 at 5:18

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

• Where is the binomial method? I think you missed that while reading the question (if you 'actually' read it). Nov 21, 2014 at 6:59
• 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. Nov 21, 2014 at 7:09
• 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. Nov 21, 2014 at 7:14

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: $$0.9^4 = (9 \cdot 10^{-1})^4 = 6561 \cdot 10^{-4} = 0.6561.$$

• If someone can evaluate that without the use of a calculator, why cant they do so for 0.9 itself Nov 21, 2014 at 7:13
• A lot of people have trouble with decimals.. Nov 21, 2014 at 23:38

$(0.9)^4=0.9\times 0.9\times 0.9\times 0.9=0.81\times 0.81=$

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

• Simple multiplication is much easier than using the binomial theorem in this case.
– JRN
Nov 21, 2014 at 1:04

It may be not what you wanted, however, it is a good mathod or tool to solve such problem.

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

• Why negative vote?
– hola
Nov 21, 2014 at 7:15