INVOLUTION.

528. Name some products made by using 2's only as

529. A power, as 4, made by using two equal factors, is a second power. A power made by using three equal factors is a third power. A power made by using four equal factors is a fourth power, etc.

530. The process of forming powers is involution.

NOTE. - The process of forming any power of a number is sometimes called raising the number to that power; the process of forming the second power is called squaring the number; the process of forming the third power is called cubing the number.

531. The second power of 2 is indicated thus, 22; the third power thus, 23; the fourth power thus, 2*, etc.

The expressions 22, 23, 24, are read, "the second power, or square, of 2," "the third power, or cube, of 2," "the fourth power of 2," respectively.

532. The number expressed by the small figure at the right is called the index or exponent of the power.

533. TO FIND ANY POWER OF A NUMBER.

ILL. EX. Find the second power or square of 100. OPERATION. 1002 100 x 100 = 10000. Ans. 10,000.

EXAMPLES.

1. Find and commit to memory the second power of each of the integral numbers from 1 to 12 inclusive. Ans. 1, 4, 9, 16, etc.

2. Find and commit to memory the third power of each of the integral numbers from 1 to 12 inclusive. Ans. 1, 8, 27, 64, etc.

Find the powers of the following numbers, as indicated by the exponents:

15. What is the difference between the square and the cube of 64 ? 16. What is the amount, at compound interest, of $1.06 for 5 years, at 6%? (Raise 1.06 to the 6th power.) Ans. $1.418519.

EVOLUTION.

534. Name one of the two equal factors of 9; one of the three equal factors of 8.

One of the equal factors of which a power is composed is a root.

535. One of the two equal factors of a second power is called a second or square root. One of the three equal factors of a third power is a third or cube root. One of the four equal factors of a fourth power is a fourth root,

etc.

536. The process of finding the root of a power is evolution.

NOTE.

The process of finding the root of a power is sometimes called extracting the root.

537. The second or square root of 64 is indicated thus, √64; the third or cube root thus, 64, etc.

In the expressions 42, 82,

162, the character denotes a root, and is called the radical sign. The radical sign, when used alone, denotes the second or square root; denotes the third or cube root; ✔ denotes the fourth root.

The number expressed by the small figure at the left of the radical sign is called the index of the root.

SECOND OR SQUARE ROOT.

538. ILL. Ex. Find the square root of 1156.

NOTE. From the definition of the square root (Art. 535), it follows, that to find the square root of 156 is to find one of its two equal factors.

539. Before attempting to find the square root of a number we will ascertain in what part of the power the square of the terms or different orders of units of the root may be found.

The second power or square of

1, a unit of the lowest order of integers,

next higher order of integers, 103, “

12, is

1.

10,

66

100.

1003, " 1000o, 66 1000000.

10000.

From the above it will be seen that the second power of a unit of any order equals 100 of the second power of a unit of the next lower order, and hence that it must be expressed two places at the left of the expression for the power of a unit of the next lower order.

540. From the above illustrations it will be seen, that the first two figures at the right of the expression for the second power of an integral number will express no part of the second power of the root above units; that the next two figures will express no part of the

second power of the root above tens; that the next two figures will express no part of the second power of the root above hundreds, and

so on.

Hence if we place a dot over the units' figure of the expression for any second power, and a dot over every alternate figure from the place of units, we shall indicate the part of the number in which the square of the units of the different orders of the root are expressed.

Then to find the number of terms or orders of units in the square root of 1156, we place a dot over each alternate figure in the expression, beginning at the units' place, thus; 1156, and having two dots, we know that there will be two terms in the root, and that the number expressed by the left-hand group contains the square of the tens of the root.

541. We will now raise a number consisting of two terms, 34 for example, to its second power, that we may learn of what parts the second power is composed.

1156, is made up

of (1) 302, or the square of the tens; (2). 30 × 4 + 30 × 4, or two products of the tens × the units; (3) 4a, or the square of the units.

In a similar manner it may be shown that the second power of any root that consists of tens with units contains (1.) The square of the tens; (2.) two products of the tens X the units; (3.) the square of the units.

Which may be expressed thus:

tens + 2 (tens X units) + units.

542. By the operation, Art. 541, we see that the square of the tens of the root can be expressed in no place lower than the hundreds’ of the power; that the 2 products of the tens X the units can be expressed in no place lower than the tens', and that the square of the units can be expressed in no place lower than the units'.

543. We are now prepared to extract the square root of 1156, which we do by taking out of the power the same partial products which were used to form it.

As the square of the tens is expressed in no place lower than the hundreds', the 11 (hunds.) must contain the square of the tens.

The greatest square contained in 11 (hunds.) is 9 (hunds.), the square root of which is 3 (tens). This we express as the first term or tens of the root.

Taking the square of 3 (tens) = 9 (hunds.) out of 11 (hunds.), there remain 2 (hunds.), which we unite with 5 (tens) of the power, making 25 (tens).

Now as the second part of the power, "2 (tens X units)," can be expressed in no place lower than the tens', the 25 (tens) must contain a product of which the tens X 2 is one factor, and the units of the root the other factor. (3 tens) X 2: 6 (tens); dividing 25 (tens) by 6 (tens), we find 4 units to be the other factor, and hence the units of the root.

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Taking out of 25 (tens) 6 (tens) × 4 24 (tens), we have 1 (ten) left, which united with 6 (units) equals 16 units, which must contain the square of 4 units. Taking out of 16 the square of 4, — 16, nothing remains; therefore the square root of 1156 is 34.

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