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

Exercises

1. Read the following words paying attention to the pronunciation:

inverse, learn, perfect, order, for, opposite, not, must, number, thus.

2. Form Participles using the following verbs:

to square, to use, to raise, to multiply, to find, to check, to give, to begin, to obtain, to get, to take, to be.

3. Make up sentences of your own using the words and expressions given below:

to raise to power, to obtain the quantity, to square the number, to take an arithmetic square root, to use a table may be checked, conclude.

IV. Answer the following questions:

1. What operation should be performed to square a number? 2. What is a perfect square? 3. What do we do to get the square root of a number? 4. What is the process of finding a root called? 5. How do we check the accuracy of a root?

V. Translate into Russian:

Tables of squares are used by architects and engineers in working with squares of number. If you have a table of squares, you can find the approximate square root of any number. Sometimes it is not easy to find a square root by

inspection. If a table of squares is not at hand another method may be used.

5. Translate into English:

Чтобы возвести в квадрат число, надо умножить это число на самое себя. Извлечение квадратного корня — это действие обратное возведению в квадрат. Чтобы получить квадратный корень числа, мы можем пользоваться специальной таблицей. Правильность извлечения квадратного корня можно проверить, возведя в квадрат подкоренное

выражение; если получится данное число, то корень найден правильно.

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