Systematic Treatise on Arithmetic, Vol. 5

The wonder of these generating rates is now illustrated through the diagram for the fifth power, which serves as a ladder A foundational guide or framework. for determining the side-rates side-rates (lianlü): the intermediate binomial coefficients used in the steps of extracting a root..

Furthermore, if we examine the square shape, it is like a square field. Multiplying the side by itself yields the area of the square; this is the:

○ The cubic shape is like the form of a die. Multiplying the square side by itself yields the square area, and then multiplying that by the height yields the cubic volume; this is the second-power The cube ($x^3$); the root is multiplied twice..

○ The third-power involves multiplying the square side by itself to get the square area, then multiplying by the height to get the cubic volume, and then multiplying by the side once more to obtain the third-power product. Therefore, it is called the "three-multiplied-power" The fourth power ($x^4$)..

○ As for the shapes of even higher powers, one cannot know what their physical forms look like; they are simply used to derive numbers. Whether one calculates the tenth-power or more than thirty powers, these are all instances where the ancient sages captured the wonder of generating rates to clarify the formal laws of root extraction root extraction (kaifang): the process of finding the side of a square, cube, or higher-order power from a given area or volume.. These methods must not be discarded.