In the 2nd example, 3 is the root, and 27 the 3rd power or cube of 3. The first power of a number is the number itself.

Q. If a number be multiplied by itself once, what is the product called? If it be multiplied by itself twice, what is the product called? What does the term power mean? What is the root?

§ 178. Involution teaches the method of finding the powers of numbers.

The number which designates the power to which the root is to be raised, is called the index or exponent of the power. It is generally written on the right, and a little above the root. Thus 42 expresses the second power of 4, or that 4 is to be multiplied by itself once: hence, 42=4x4=16.

For the same reason 33 denotes that 3 is to be raised to the 3rd power, or cubed: hence

33=3x3x3=27: we may therefore write,

4=4

42=4x4=16

4=4x4x4=64

the 1st power of 4.

the 2nd power of 4.

the 3rd power of 4.

4*=4x4x4x4=256 the 4th power of 4.

45=4x4x4x4×4=1024 the 5th power of 4.

Q. What is Involution? What is the number called which designates the power? Where is it written?

Hence, to raise a number to any power, we have the following

RULE.

Multiply the number continually by itself as many times less 1 as there are units in the exponent: the last product will be the power sought.

5. What is the 5th power of 9?
6. What is the cube of 1?
7. What is the square of ?
8. What is the cube of,1?

9. What is the cube of?
10. What is the square of ,01 ?
11. What is the square of 2,04?
12. What is the 5th power of 10?
13. What is the cube of 21?

Q. How do you raise a number to any power?

EVOLUTION.

§ 179. We have seen (§ 178,) that Involution teaches how to find the power when the root is given. Evolution is the reverse of Involution: it teaches how to find the root when the power is known. The root is that number which being multiplied by itself a certain number of times will produce the given power.

The square root of a number is that number which being multiplied by itself once will produce the given number.

The cube root of a number is that number which being multiplied by itself twice will produce the given number. For example, 6 is the square root of 36; because 6×6 =36; and 3 is the cube root of 27, because 3×3×3= 27. The sign placed before a number denotes that its square root is to be extracted. Thus, 36=6. The sign is called the sign of the square root.

When we wish to express that the cube root is to be extracted, we place the figure 3 over the sign of the square root: thus, 8=2 and 27=3.

Q. What is Evolution? What does it teach? What is the square .root of a number? What is the cube root of a number? Make the sign denoting the square root? How do you denote the cube root?

EXTRACTION OF THE SQUARE ROOT.

§ 180. To extract the square root of a number, is to find a number which being multiplied by itself once, will produce the given number. Thus

Before proceeding to explain the rule for extracting the square root, let us first see how the squares of numbers are formed.

The first ten numbers are

1

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Roots. 1 4 9 16 25 36 49 64 81 100 Squares. the numbers in the second line are the squares of those in the first: and the numbers in the first line are the square roots of the corresponding numbers of the second.

Now, it is evident that, the square of a number expressed by a single figure will not contain any figure of a higher order than tens. And also, that if a number contains three figures its root must contain tens and units.

The numbers 1, 4, 9, &c. of the second line, are called perfect squares, because they have exact roots.

Let us now see how the square of any number may be formed: say the number 36. This number is made up of 3 tens or 30, and 6 units. Let the line AB represent the 3 tens or 30, and BC the six units.

Let AD be a square on H AC, and AE a square on the tens line AB.

Then ED will be a square on the unit line 6, and the rectangle EF will be the product of HE which is equal to the tens line, by IE which is equal to the unit line.

Also, A

F

30

I

D

30

6

6

180

36

the rectangle BK will be the product of EB which is equal to the tens line, by the unit line BC. But the whole square on AC is made up of the square AE, the two rectangles FE and EC, and the square ED: Hence

The square of two figures is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.

Let it now be required to extract the square root of

1296.

Since the number contains more than two places, its root will contain tens and units. But as the square of one ten is one hundred, it follows that the ten's place quired root must be found in the figures on the Hence, we point off the number into periods of two figures each.

of the releft of 96. 12 96(36

9

66)396

396

We next find the greatest square contained in 12, which is 3 tens or 30. We then square 3 tens which gives 9 hundred, and then place 9 under the hundred's place, and subtract.

This takes away the square AE and leaves the two rectangles FE and BK, together with the square ED on the unit line.

H

30

Now, since tens multiplied by units will give at least tens in the product, it follows that the area of the two rectangles FE and EC must be expressed by the figures at the left of the unit's place A

F

6, which figures may also express a part of the square ED. If, then, we divide the figures 39, at the left of 6, by twice the tens, that is, by twice AB or BE, the quotient will be BC or EK, the unit place of the root.

Then, placing BC or 6, in the root, and also in the divisor, and then multiplying the whole divisor 66 by 6, we obtain for a product the two rectangles, FE and EC together with the square ED.

Hence, the square root 1296 is 36; or, in other words 36 is the side of a square whose area is 1296.

Hence we have

CASE 1.

§ 181. To extract the square root of a whole number.

RULE.

1. Point off the given number into periods of two figures each, counted from the right, by setting a dot over the place of units, another over the place of hundreds, and so on.

II. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of the root from the first period, and to the remainder bring down the second period for a dividend.

III. Double the root already found and place it on the left for a divisor. Seek how many times the divisor is contained in the dividend, exclusive of the right hand figure, and place the figure in the root and also at the right of the divisor.

IV. Multiply the divisor, thus augmented, by the last figure of the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. But if the product should exceed the dividend, diminish the last figure of the root.

V. Double the whole root already found, for a new divisor, and continue the operation as before, until all the periods are brought down.

Q. What is required when we wish to extract the square root of a number? What is the greatest square of a single figure? What is the highest order of units that can be derived from the square of a single figure? How many perfect squares are there among the numbers that are less than one hundred?

What is the square of a number expressed by two figures equal to? In what places of figures will the square of the tens be found? In what places will the product of the tens by the units be found? What is the first step in extracting the square root of numbers? What the second? What the third? What the fourth? What the fifth? Give the entire rule.