How do you work out $\sqrt[4]{16^3}$ without a calculator.

Question

$$\sqrt[4]{16^3}$$

I just don't know what to do when I get to $4096$. The original equation was $16^{3/4}$.

Answer 1

Let $\displaystyle a^4=16\implies a^2=\pm4$

$\displaystyle\implies a=\pm2,\pm2i$

the principal value of $a$ being $2$

$$16^{\frac34}=\left(a^4\right)^{\frac34}=a^3$$

Answer 2

$$16^{3/4} = (2^4)^{3/4} = 2^{4\cdot(3/4)} = 2^3 = 8.$$

First we note that $16 = 2^4$, after that we use that $(a^n)^m = a^{n\cdot m}$ (this holds in general for all non-negative real $a$).

Answer 3

If you recognize that $16^{\frac{1}{4}}=2$ (since $2^4=2\cdot2\cdot2\cdot2=4\cdot4=16$), then $$16^{3/4}=(16^{\frac{1}{4}})^{3}=(2)^3=\boxed{8}$$

Answer 4

The really basic way to do this:

$$\begin{align}(16^3)^{1/4} &= ((2^4)^3)^{1/4}\\ &=2^{4 \cdot 3 \cdot 1/4}\\ &=2^3\\ &= 8 \end{align}$$

Note that -8 is an equally valid solution since the power is even. If the answer is meant to be "easy" then usually the solution is found by expressing things in terms of the prime factors.

As an aside, it gets more interesting when the answer is not an integer (in other words, when the number you start with is not a perfect square). There is a neat trick (which they no longer teach in most schools) for getting the square root of any number to arbitrary precision (limited by your patience, accuracy, size of your paper and the sharpness of your pencil). It looks a bit like long division. See "Method 2" of http://www.wikihow.com/Calculate-a-Square-Root-by-Hand

It can work for any root that is a power of 2 (2, 4, 8...) - you just have to take the root of the root, etc.