Squares and square roots

To square a number¹, you have learned, you must multiply that number by itself. The square root of a number is just the opposite. When you find the square root of a number, you are finding what number multiplied by itself gives you the number you began with². The sign for the square root is √. Thus, the square root of 25 is represented by √25. 25 is a perfect square. That is, a whole number (5) multiplied by itself will give you 25. Most numbers are not perfect

squares. In that case, to get the square root of a number we may either find it by taking an arithmetic square root or by using a table.

The process of finding a root is known as evolution; it is the inverse of involution, because by the aid of this process we try to find that which is given only when raising a number to a power (viz. the base of the power), while the data given is just what is sought for³ raising a number to a power (viz. the power itself). Therefore the accuracy of the root taken may always be checked by raising the number to the power⁴. For instance, in order to check the equality: ³√125=5, it is sufficient to cube 5; obtaining the quantity under the radical

sign, we conclude that the cube root of 125 has been found correctly.

Notes:

¹ to square a number — чтобы возвести число в квадрат

² the number you began with — зд. исходное число

³ what is sought for — искомое

⁴ by raising the number to the power — возведением числа в степень