A pinch of history

These ideas are very old. The first known lists of perfect squares and perfect cubes were made by the Babylonians as far back as 1700 BCE. They were found on clay tablets, and were used to look up square roots and cube roots quickly for land measurement, building design and other work needing geometry.

In ancient Sanskrit works, the word varga meant both the square figure (or its area) and the square power: the same double meaning our word 'square' still has. Ghana meant both the solid cube and a number multiplied by itself three times. The fourth power was varga-varga, the square of the square. These words were used in India from at least the third century BCE.

Aryabhata, writing in 499 CE, put it exactly: 'A square figure of four equal sides and the number representing its area are called varga. The product of two equal quantities is also called varga.' So the name of the power came from the picture of the shape.

Now a stranger question: why do we call √ a root, like the root of a plant?

Because in ancient India the Sanskrit word mula, meaning the root of a plant (and also basis, cause or origin), was used for exactly this. Varga-mula, the origin of the square, meant square root; ghana-mula meant cube root.

'This use of mula for the mathematical concept of root was subsequently emulated in Arabic and Latin through their corresponding words for the root of a plant: jidhr and radix respectively.' The word mula has been used this way in India since at least the first century BCE. So every time you say 'square root', you're using a very old Indian picture: the number a square grows from.

Another Sanskrit word for the same idea was pada: foot, basis, cause, origin. Brahmagupta explained it in 628 CE: 'The pada (root) of a krti (square) is that of which it is a square.'