Better methods to approximate $2^{2\over 3}$

Question

Recently while solving a problem on thermodynamics I ended up with $2^{2\over 3}$ .

Now the problem was on a test where no calculators were allowed and answer was required upto $2$ decimal digits.

I then resorted to binomial theorem for help (for $x\lt 1$) $$\left. \begin{array} { l } { ( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) x ^ { 2 } } { 2 ! } + \frac { n ( n - 1 ) ( n - 2 ) } { 3 ! } x ^ { 3 } + \ldots \ldots + \frac { n ( n - 1 ) \ldots \ldots ( n - r + 1 ) } { r ! } x ^ { r } \ldots } \\ \end{array} \right.\text{upto}\, \, \infty$$ So the original problem can be written as :

$$2^{2\over 3}=4^{1\over 3}=(8-4)^{1\over 3}=2\left(1-\frac{1}{2}\right)^{1\over 3}$$

Now after evaluation first $3$ terms I ended up with $\left(2-\frac{1}{3}-\frac{1}{18}\right) \approx 1.61$ but the correct answer was $1.59$.

Also the average time you got per question was around $2$ minutes and I had already used more than half of it so I did not think of calculating more terms.

Now I am looking for a method which can help me evaluate $2^{2\over 3}$ faster and more precisely.

Can anyone please help me with this ?

Answer 1

If you are around powers of 2 a lot (as computer scientists are, for example), then you can recognize $4.096=2^{12}/10^3=(2^4/10)^3=1.6^3$, and then $4^{1/3} = 1.6(1-\frac{0.096}{4.096})^{1/3}$ is much better approximated by its power series. Indeed, the approximation $1.6(1-\frac13\frac{0.096}{4.096}) \approx 1.6(1-\frac13\frac{0.1}4) = 1.6 - \frac1{75}$ is already within $0.05\%$ of the true answer $\approx 1.5874$.

Answer 2

How about using some Calculus? We got to compute $\sqrt[3]{4}$, so let $y=f(x) =\sqrt[3]{x}$. Now, we know that $f(8)= 2$ and $$ \frac{dy}{dx}= \frac{1}{3} \frac{1}{x^{2/3}}$$

$$\Delta x = 4 - 8 = -4 \\ \Delta y = 1/3 \frac{1}{x^{2/3}} \Delta x \\ \Delta y = \frac{1}{3 \times 4} (-4) \\ \Delta y = -0.3333 $$

So, $f(4) = 2 - 0.3333= 1.667$.

I know my answer is very far from original one, and to be honest it looks to me that they have used the calculator for the correct answer, I added this method because you've mentioned the time limit and wanted something quick.

Answer 3

Consider that you look for the zero of function $$f(x)=x^3-4$$ We have $f(1)=-3$ and $f(2)=+4$. Using $x_0=\frac 32$ (the midpoint), the first iterate of Newton method is $\frac{43}{27} \approx 1.59259$ while the exact value is $1.58740$.

The answer is then $1.59$.

Edit

You do not need to know about Newton method since what I did is just linearization (or Taylor series) $$f(x)=f(x_0)+f'(x_0) (x-x_0)+ O((x-x_0)^2$$

so for you case $$0=-\frac 58+\frac {27}4 (x-\frac 32)=\frac{27 }{4}x-\frac{43}{4}\implies x=\frac{43}{27}$$

Answer 4

$\displaystyle\,\sqrt[\large 3]{\, 4\, } = \sqrt[\large 3]{512 - 12 \over 125} = {8 \over 5}\left(1 - {3 \over 128}\right)^{1/3} \approx 1.6 \times \left(1 - {1 \over 128}\right) = 1.587\color{red}{5}$

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In general ( expanding the above binomial ), with $\displaystyle{8 \over 5}\sum_{k = 0}^{n}{1/3 \choose k} \left(\,{-\,{1 \over 128}}\,\right)^{k}$: $$ {\Large\,\sqrt[\large 3]{\, 4\, }\,}\ \approx\ \left\{\begin{array}{lll} {\displaystyle 1.\color{red}{6}} & {\displaystyle n = } & {\displaystyle 0} \\ {\displaystyle 1.587\color{red}{5\ldots}} & {\displaystyle n = } & {\displaystyle 1} \\ {\displaystyle 1.58740\color{red}{2\ldots}} & {\displaystyle n = } & {\displaystyle 2} \\ {\displaystyle 1.5874010\color{red}{7\ldots}} & {\displaystyle n = } & {\displaystyle 3} \\ {\displaystyle 1.58740105\color{red}{2\ldots}} & {\displaystyle n = } & {\displaystyle 4} \\ {\displaystyle 1.5874010519\color{red}{7\ldots}} & {\displaystyle n = } & {\displaystyle 5} \\ {\displaystyle 1.587401051968\color{red}{3\ldots}} & {\displaystyle n = } & {\displaystyle 6} \end{array}\right. $$

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Result for some machine software: $$ 1.587401051968199474751705639272308260391493327899853009808\ldots $$

Answer 5

Perhaps more elaboration on how to use Newton's method, since I personally find it very easy to use to find the first few digits of $n$th roots using only basic calculations.

From the binomial expansion, we know that

$$(x+\Delta x)^3=x^3+3x^2\Delta x+\mathcal O((\Delta x)^2)$$

Our goal is essentially finding the $\Delta x$ that fixes $x^3$ so that $(x+\Delta x)^3$ is closer to $4$. In this way, we can find $\sqrt[3]4$.

Setting it equal and solving, we end up with

$$x^3+3x^2\Delta x=4\implies\Delta x=\frac{4-x^3}{3x^2}=\dfrac{\dfrac4{x^2}-x}3$$

Adding $x$ to both sides to find our new $x$ gives us

$$x+\Delta x=\frac{4+2x^3}{3x^2}=\dfrac{\dfrac4{x^2}+2x}3$$

Starting with values of $x$ closer to $\sqrt[3]4$ such as the $1.61$ you found will lead to faster correction, but for example purposes we start far away at $x=1$, which makes the next estimate become

$$x+\Delta x=\frac{4+2}3=2$$

Substituting $x=2$ in now gives us

$$x+\Delta x=\frac{4+16}{12}=\frac53\simeq1.66$$

The next estimate is a tad messier but still manageable:

$$x+\Delta x=\frac{4+2\times\frac{125}{27}}{3\times\frac{25}9}=\frac{4\times27+2\times125}{3\times3\times25}=\frac{108+250}{225}=\frac{358}{225}\simeq1.591$$

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In general you will find that

$$x+\Delta x=\frac{a+(n-1)x^n}{nx^{n-1}}$$

for computing $\sqrt[n]a$, which works out to give the first few digits fairly fast and easily provided a good initial estimate (as you had done) and that $n$ is not too large.