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

Question

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 $(1-0.1)^4$.

Answer 1

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

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

Answer 2

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

Answer 3

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

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

Answer 4

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