Easy exponentiation method

Question

Is there a simple way of solving, say, $x^{3/2}$?

For example, one way of solving $16^{3/2}$ is to calculate the square root of $16^3$, but I was wondering if there is a simpler mental trick for doing this that generalizes to all possible exponentiation.

Answer 1

One of the possible ways is to apply an exponential rule:

$$x^{\frac{3}{2}}=x\cdot\sqrt{x}$$

Example:

$$16^{\frac{3}{2}}=16\cdot\sqrt{16}=16\cdot4=64$$

Answer 2

If you could see that $16=4^2$ then you could do this

$$\left(4^2\right)^{3/2}=4^3$$

If you can see such a number that would be the fastest method, another method would be the one rae306 mentioned.

Answer 3

Following Troy Woo and Travis comments, let us make it more general : you want to know $x$ such that $$x^a=b$$ and you know already a solution for a given $a$ and a value close to $b$.

Newton method is quite simple since, starting at your known estimate $x_0$, the first iterate will be $$x_1=x_0-\frac{x_0^{1-a} \left(x_0^a-b\right)}{a}$$

Let us apply it to $a=\frac{3}{2}$, $b=10$ and $x_0=4$. This gives $$x_1=\frac{14}{3} \simeq 4.66667$$ while the exact solution is $4.64159$