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2206                                                                ARITHMETIC

The product of three like factors is the 3rd power, or cube of that number.

27 is the cube of 3 (3X3 X3 = 27). 64 is the cube of 4 (4 X 4 X 4 = 64).

The product of four like factors is the 4th power; of five like factors is the 5th power, etc.

The exponent is a small figure written a little above the number at the right, and it tells how many times the number is taken as a factor.

62 = 6X6      73 = 7X7X7       5* = 5X 5X5X5 The root of a number is one of the equal factors of the number.

The square root of a number is one of the two equal factors of the number.

The square root of 81 is 9; of T% is f.

The cube root of a number is one of the three equal factors of the number.

The cube root of 125 is 5 • of ^\ is f.

The symbol V? called the radical sign, written before a number, shows that the square root of the number is to be taken. y~49" = square root of 49 = 7.

When some root other than the square root is desired, a small number, called the index, is written a little above and before the radical sign, to indicate what root is to be found. 3\/64"= cube root of 64 = 4.

16                             32 = 9

452 = 2025                        212 = 441

The square of a number contains twice as many figures, or one less than twice as many figures, as the root.

Therefore, for every one or two figures in the power there will be one figure in the root, and for every three or four figures in the power there will be two figures in the root. These groups of two figures are called periods.

Thus in every root there will be as many figures as there are periods, or fraction of a period, in the power.

Since 45 = 40 + 5, the square of 45 may be obtained as follows :

4Q T 5                                    The square of any number made up of tens

402~_j_ (40 v 5)                 and nnits will contain the square of the tens +

(40 X 5) 4- 52         twice the product of the tens by the units + the

- — - — - -      square of the units,

402 + 2 (40X5) +52        H