A Bit of Algebra

In this blog post we're going to look at finding the real positive x such that: \[x^{x^3} = \frac{1}{2^{1/6}} \hspace{10 mm} (Equation 1)\]
By all means, do have a go at solving it, before reading our solution below.

The first, slightly non-obvious step is to raise both sides of the equation to the power of 3: \[ \left( x^{x^3} \right)^3 = \left( \frac{1}{2^{1/6}} \right)^3 \] Rearranging that a bit, we have: \[ x^{3x^3} = \frac{1}{2^{3/6}} \] And so: \[ \left( x^3 \right) ^{x^3} = \left( \frac{1}{2} \right) ^ {1/2} \hspace{10 mm} (Equation 2) \] If \[ a^a = b^b \] then we have a solution: \[a=b\]
So, returning to Equation 2, we have: \[x^3=1/2 \] And hence: \[x=\frac{1}{2^{1/3}} \] So, we have a solution to equation 1. But are there any others?
In fact, it is possible have \[ a \neq b \] when \[ a^a = b^b \] For example: \[ \left( \frac{1}{4} \right)^{1/4} = \left( \frac{1}{2} \right)^{1/2} \] So, returning to equation 2, we can see that another solution for x is when: \[ x^3 = \frac{1}{4} \] And hence: \[ x = \frac{1}{2^{2/3}} \] So, our solutions for x are: \[ \frac{1}{2^{1/3}} \hspace{4 mm} and \hspace{4 mm} \frac{1}{2^{2/3}} \] But are there other solutions?
If we let \[ z=x^3 \] then, returning to equation 2 we have: \[ z^z= \frac{1} {\sqrt{2}} \hspace{10 mm} \] We can plot \(z^z \) for positive z:

We can see that \(z^z\) decreases in the range 0 to 1/e, after that it increases and keeps on increasing. So for positive z, we will have a maximum of two solutions to \( z^z = k\), where k is some constant. We have two solutions already, so we now know there aren't any more,
unless of course we want to investigate negative or complex solutions...

Catching Raindrops

Suppose we know the area of a country and the average annual rainfall. And then suppose that we also know the population and the daily water consumption, then we can work out what proportion of drops of rain that fall need to be captured for use by people.
We convert the daily consuption to annual and then compare the volume of water that is consumed with the volume of water that falls as rain.

Inputs
Population		people
Average Water Consumption		litres per person per day
Land Area		square kilometers
Rainfall		meters per year
Result
Proportion of raindrops to catch		%

(If you change one of the inputs above and hit enter, then the result will be updated.)

Singapore has approximately a population of 5.9 million people, they use 141 litres of water per person per day. The land area is small: 729 sq km, their annual rainfall is about 2.2 meters. So if they captured 19% of the raindrops then they'd meet their water needs.
On the other hand, Ireland has a slightly smaller population but much larger land area and so they only need capture a much lower proportion of their raindrops.