Charles Davies.

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The third power is the product when the root is taken 3
times as a factor :

The fourth power, when it is taken 4 times :
The fifth power, when it is taken 5 times, &c.

288. The number denoting how many times the root is
taken as a factor, is called the exponent of the power. It is
written a little at the right and over the root : thus, if the
equal factor or root is 4,

4= 4 the 1st power of 4.

4 2 4x4= 16 the 2d power of 4.

43 _4 X 4 X 4 64 the 3d power of 4.

4 4 =4: x 4 x 4 x 4 = 256 the 4th power of 4.

45 .-4x4x4x4x4 1024 the 5th power of 4.

INVOLUTION is the process of finding the powers of 'number 's.

NOTES. 1. There are three things connected with every power :
1st, The root ; 2d, The exponent ; and 3d, The power or result of
the multiplication.

2 In finding a power, the root is always the 1st power; hence,
the number of multiplications is 1 less than the exponent;

RULE. Multiply the number by itself as many times less
1 as there are units in the exponent, and the last product
will be the power.

EXAMPLES.

Find the powers of tne following numbers :



1. Square of 1.

2. Square of J.

3. Cube of |.

4. Square of f .

5. Square of 9.

6. Cube of 12

1. 3d power of 125.

8. 3d power of 16

9. 4th power of 9.



10. 5th power of 16.

11. 6th power of 20.

12. 2d power of 225

13. Square of 2167.

14. Cube of 321

15. 4th power of 215.

16. 5th power of 906.

17. 6th power of 9.

18. Square of 36049.



EVOLUTION. 277

EVOLUTION.

289, EVOLUTION is the process of finding the factor when
we know the power.

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

The cube root of a number is the factor which multiplied
by itself twice will produce the number.

Thus, 6 is the square root of 36, because 6 x 6=36 ; and
3 is the cube root of 27, because 3 x 3 x 3=27.

The sign V is called the radical sign. When placed be-
fore a number it denotes that its square root is to be ex-
tracted. Thus, 1/36 = 6.

We denote the cube root by the same sign by writing 3
over it : thus, v^ denotes the cube root of 27, which is
equal to 3. The small figure 3, placed over the radical, is
called the index of the root.

'EXTRACTION OF THE SQUARE ROOT.

290. The square root of a number is a factor which mul-
tiplied by itself once will produce the number. To extract
the square root is to find this factor* The first ten numbers
and their squares are

1, 2, 3, 4, 5, 6, Y, 8, 9, 10.
1, 4, 9, 16, 25, 36, 49, 64, 81. 100.
The numbers in the first line are the square roots of those
in the second. The numbers 1, 4, 9, 16, 25, 36, &c.
having exact factors, are called perfect squares.

A perfect square is a number which has two exact factors

NOTE. The square root of a number less than 100 will be less
than 10, while the square root of a number greater than 100 will
be greater than 10.

287. What is a power ? What is the root of a power? What is the
first power ? What is the second power ? The third power ?

288. What is the exponent of the power ? How is it written ? What
is Involution ? How many things are connected with every power ?
How do you find the power of a number ?

289. What is Evolution? What is the square root of a number?
What is the cube root of a number ? How do you denote the square
root of a number ? How the cube root ?



278



EXTRACTION OF THE SQUARE ROOT.



291. What is the square of 36=3 tens + 6 units?



ANALYSIS. 36=3 tens+6 units, is first
to be taken 6 units' time, giving 6 2 +3 x 6 :
then taking it 3 tens' times, we have
3 x 6+3 2 , and the sum is 3 2 +2(3 x 6)+6 2 :
that is,



3 + 6
3 + 6

3x6 + 6*
3 2 +3x6



3 2 +2(3x6)+6

The square of a number is equal to the square of the tens,
plus twice the product of the tens by the units, plus the square
of the units.

The same may be shown by the figure :

Let the line AB re- F 30

present the 3 tens or 30,
and BC the six units.

Let AD be a square
on AC, and AE a square
on the ten's 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 ten's line, by IE,
which is equal to the
unit line Also, the
rectangle BK will be the
product of EB, which is
equal to the ten's line, by
the unit line B C. But the whole square on AC is made up of
the square AE, the two rectangles FE and EC, and the square
ED.

1. Let it now be required to extract the square root of
1296.

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



30
6

180


6
6
36


30 E
900 + 180 + 180 + 36=1296.




30
30


30
6


900


180



30



C



290. What is the square root of a, number ? What are perfect
squares ? How many are there between 1 and 100 ?

291. Into what parts may a number be decomposed? When so de-
composed, what is its square equal to ?



EXTRACTION OF THE SQUARE ROOT. 279

We next find the greatest square contained in OPERATION.
12, which is 3 tens or 30. We then square 3 1296(36

tens which gives 9 hundred, and then place 9 un- ~

der the hundreds' place, and subtract , this takes
away the square of the tens, and leaves 396, 66)396

which is twice the product of the tens by the units 395

plus the square of the units.

If now, we double the divisor and then divide this remainder,
exclusive of the right hand figure, (since that figure cannot enter
into the product of the tens by the units) by it, the quotient will
be the units figure of the root. If we annex this figure to the
augmented divisor, and then multiply the whole divisor thus in-
creased by it, the product will be twice the tens by the units plus
the square of the units ; and hence, we have found both figures of
the root.

This process may also be illustrated by the figure.

Subtracting the square of the tens is taking away the square
AE and leaves the two rectangles FE and BK, together with the
Bquare ED on the unit line.

The two rectangles FE and BK*representing the product of units
by tens, can be expressed by no figures less than tens.

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 BG or EK
the unit of the root.

Then, placing BC or G, in the root, and also annexing it to the
divisor doubled, and then multiplying the whole divisor 66 by 6,
we obtain the two rectangles FE and CE, together with the
equare ED.

292. Hence, for the extraction of the square root, we have
the following

RULE. I. Separate the given number into periods of two
figures each, by setting a dot over the place of units, a se-
cond over the place of hundreds, and so on for each alternate
figure at the left.

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

292. What is the first step in extracting the square root of numbers ?
What is the second? What is the third? What the fourth? What
the fifth ? Give the entire rule.



280 EXTRACTION OF THE SQUARE ROOT.

III. Double the root thus found for a trial divisor and
place it on the left of the dividend. Find how many
times the trial divisor is contained in the dividend, exclu-
sive of the right-hand figure, and place the quotient in the
root and also annex it to the divisor.

IY. Multiply the divisor thus increased, by the last figure
of the root ; subtract the product from the dividend, and to
the remainder bring down the next period for a new divi-
dend.

Y. Double the ivhole root thus found, for a new trial di-
visor, and continue the operation as before, until all the
periods are brought down.

EXAMPLES.

1. What is the square root of 263169 ?

OPERATION.

ANALYSIS. We first place a dot over the a o'i A 6 / K i Q

9, making the right-hand period 69. We
then put a dot over the 1 and also over the
6, making three periods. 101)131

The greatest perfect square in 26 is 25, AI

the root of which is 5, Placing 5 in the



root, subtracting its square from 26, and 1023)3069
bringing down the next period 31, we have 3069

131 for a dividend, and by doubling the

root we have 10 for a trial divisor. Now, 10 is contained in 13,
1 time. Place 1 both in the root and in the divisor : then multi-
ply 101 by 1 ; subtract the product and bring down the next period.
We must now double the whole root 51 for a new trial divisor ;
or we may take the first divisor after having doubled the last
figure 1 ; then dividing, we obtain 3, the third figure of the root.

NOTE. 1. The left-hand period may contain but one figure;
each of the others will contain two.

2. If any trial divisor is greater than its dividend, the corres
ponding quotient figure will be a cipher.

3. If the product of the divisor by any figure of the root exceeds
the corresponding dividend, the quotient figure is too large and
must be diminished.

4. There will be as many figures in the root as there are periods
in the given number.

5. If the given number is not a perfect square there will be a
remainder after all the periods are brought down. In this case,
periods of ciphers may be annexed, forming new periods, each of
which will give one decimal place in the root.



EXTRACTION OF THE SQUARE ROOT.



281



What is the square root of 36729 : OPERATION.

3 67 29(191.64 +
1



In this example there are two
periods of decimals, which give two
places of decimals in the root.



29)267
261

381)629
381



3826)24800
22956



38324)184400
153296
31104 Hem.



293. To extract the square root of a fraction.



1. What is the square root of .5 ?



NOTE. We first annex one cipher to
make even decimal places. We then ex-
tract the root of the first period : to the
remainder we annex two ciphers, forming
a new period, and so on.



OPERATION.

.50(.707 +
49

140)100
000



1407)10000
9849



151 Rem.



OPERATION.



2. What is the square root of ?

NOTE. The square root of a fraction
is equal to the square root of the numerator
divided by the square root of the denomi-
nator.



3. What is the square root of J ?

NOTE. When the terms are not per-
fect squares, reduce the common fraction | = . 7 5 ;
to a decimal fraction, and then extract x /sZr v /VcT_
the square root of the decimal. 5 *&



OPERATION.



293. How do you extract the square root of a decimal fraction ?
ef a common fraction ?



How



282



SQUARE ROOT.



RULE. I. If ike fraction is a decimal, point off the
periods from the decimal point to the right, annexing ci-
phers if necessary, so that each period shall contain two
places, and then extractJhe root as in integral numbers.

II. If the fraction is a common fraction, and its terms
perfect squares, extract the square root of the numerator and
denominator separately ; if they are not perfect squares, re-
duce the fraction to a decimal, and then extract the square
root of the result.

EXAMPLES.

What are the square roots of the following numbers ?



of 3?
of 11?
of 1069 ?
of 2268741?



5. of 7596796?



of 36372961?
of 22071204?
of 3271.4207?
of 4795.25731?



10. of 4.372594?



11. of .0025?

12. of .00032754?

13. of .00103041?

14. of 4.426816?

15. of8f ?

16. of 9J?

17. of^?

18. o

19. o

20. off



APPLICATIONS IN SQUARE ROOT.



294. A triangle is a plain figure which has three sides and
three angles.



If a straight line meets another straight line,
making the adjacent angles equal, each is
called a right angle ; and the lines are said
to be perpendicular to each other.

295. A right angled triangle is one
which has one right angle. In the right
angled triangle ABC, the side AC opposite
the right angle B is called the hi/pothenuse ;
the side AB the base; and the side BC
the perpendicular.




APPLICATIONS.



283



29G. In a right angled triangle the square described in
the hypothemise is equal to the sum of the squares described
in the other two sides.

Thus, if AC13 be a right
angled triangle, right an-
gled at C, then will the
large square, D, described
on the hypothenuse AB, be 1
equal to the sum of the
squares F and E described
on the sides AC and CB.
This is called the carpen-
ter's theorem. By count-
ing the small squares in the
large square D, you will
find their number equal
to that contained in the

small squares F and E. In this triangle the hypothenuse
AB = 5, AC = 4, and CB = 3. Any numbers having the
same ratie, as 5, 4 and 3, such as 10, 8 and 6 ; 20, 16 and
12, &c., will represent the sides of a right angled triangle.




1. Wishing to know the distance from A
to the top of a tower, I measured the height
of the tower and found it to be 40 feet ; also
the distance from A to B and found it 30 feet ;
what was the distance from A to C ?
30 2 = 900



BC=40; BC^40 2 ^



~ 2500



= ^2500 = 50 feet.




297. Hence, when the base and perpendicular are known
and the hypothenuse is required,



294. What is a triangle ? What is a right angle ?

295. What is a right angled triangle ? Which side is the hypothe-
nuse ?

296. In a right angled triangle what is the square on the hypothe-
nuse equal to ?



284 SQUARE ROOT.

Square the base and square the perpendicular, add the re-
sults and then extract the square root of their sum.

2. What is the length of a rafter that will reach from the
eaves to the ridge pole of a house, when the height of the
roof is 15 feet and the width of the building 40 feet ?

298. To find one side when we know the hypothenuse and
the other side.

3. The length of a ladder which will reach from the mid-
dle of a street 80 feet wide to the eves of a house, is 50 feet :
what is the height of the house ? Ans. 30 feet.

ANALYSIS Since the square of the length of the ladder is equal
to the sum of the squares of half the street and the height of the
house, the square of the length of the ladder diminished by the
square of half the street will be equal to the square of the height
of the house : hence,

Square the hypothenuse and the known side, and take the
difference ; the square root of the difference will be the other
side.

EXAMPLES.

1. If an acre of land be laid out in a square form, what
will be the length of each side in rods ?

2. What will be the length of the side of a square, in rods,
that shall contain 100 acres ?

3. A general has an army of 7225 men : how many must
be put in each line in order to place them in a square form ?

4. Two persons start from the same point ; one travels
due east 50 miles, the other due south 84 miles : how far are
they apart ?

5. What is the length, in rods, of one side of a square that
shall contain 12 acres ?

6. A company of speculators bought a tract of land for
$6724, each agreeing to pay as many dollars as there were
partners : how many partners were there ?

297. How do you find the hypothenuse when you know the base
and perpendicular ?

298. If you know the hypothenuse and one side, how do you find the
other side ?



CUBE ROOT. 285

7. A farmer wishes to set out an orchard of 3844 trees, so
that the number of rows shall be equal to the number of
trees in each row : what will be the number of trees ?

8. How many rods of fence will enclose a square field of
10 acres ?

9. If a line 150 feet long will reach from the top of a
steeple 120 feet high, to the opposite side of the street, what
is the width of the street ?

10. What is the length of a brace whose ends are each 3|
feet from the angle made by the post and beam ?

CUBE ROOT.

299. The CUBE ROOT of a number is one of three equal
factors of the number.

To extract the cube root of a number is to find a factor
which multiplied into itself twice, will produce the given
number.

Thus, 2 is the cube root of 8 ; for, 2 x 2 x 2 = 8 : and 3 is
the cube toot of 27 ; for 3 x 3 x 3 = 27.

1, 2, 3, 4, 5, 6, 7, 8, 9.

1 8 27 64 125 216 343 512 729.

The numbers in the first line are the cube roots of the
corresponding numbers of the second. The numbers of the
second line are called perfect cubes. By examining the num-
bers of the two lines we see,

1st. That the cube of units cannot give a higher order than
hundreds.

2d. That since the cube of one ten (10) is 1000 and the
cube of 9 tens (90), 81000, the cube of tens will not give a
lower denomination than thousands, nor a higher denomi-
nation than hundreds of thousands.

Hence, if a number contains more than three figures, its
cube root will contain more than one : if it contains more
than six, its root will contain more than two, and so on ;
every additional three figures giving one additional figure in
the root, and the figures which remain at the left hand,
although less than three, will also give a figure in the root,
This law explains the reason for pointing off into periods of
three figures each.



286 CUBE BOOT.

300. Let us now see how the cube of any number, as 16,
is formed. Sixteen is composed of 1 ten and 6 units, and
may be written 10 -f G. To nod the cube of 16, or of 10+6,
we must multiply the number by itself twice

To do this we place the number thus 16=10-}- 6

10+ 6

product by the units - 60+36

product by the tens -100+ 60

Square of 16 - 100+ 120 - *- 36

Multiply again by 16 - - 10+6

product by the units - 600+ 720+216

product by the tens 1000+1200+ 360

Cube of 1 6 TOOO+T800 + 1080 + 2l6

1. By examining the parts of this number it is seen that
the first part 1000 is the cube of the tens ; that is,

10x10x10=1000.

2. The second part 1800 is three times the square of the
tens multiplied by the units ; that is,

3 x (10)* x 6=3 x 100 x 6=1800.

3. The third part 1080 is three times the square of the units
multiplied by the tens ; that is,

3 x6 2 x 10=3x36x10=1080.

4. The" fourth part is the cube of the units ; that is,

6 3 =6x 6x6=210.
1. What is the cube root of the number 4096 ?

ANALYSTS. Since the number

contains more than three figures, 4 096(16

we iuaow that the root will con- 1

tain at least units and tens. ia o \1T~n 7c\ Q T R

. Separating the three right- l*X_3 = o)3 I
hand figures from the 4, we 16 3 =4 096

know that the cube of the tens
\vili be found in the 4 ; and 1 is the greatest cube in 4.

299. What is the cube root of a number ? How many perfect cubes
arc there between 1 and 1000 ? Tin,.*

800. Of how many parts is the cube of a number composed ? What
are they ?



CUBE BOOT. 287

Hence, we place the root 1 on the right, and this is the tens of
the required root. We then cube 1 and subtract the result from
4, and to the remainder we bring down the first figure of the
next period.

We have seen that the second part of the cube of 16, viz. 1800,
is three times the square of the tens multiplied by the units : and
hence, it can have no significant figure of a less denomination than
hundreds. It must, therefore, make up a part of the 30 hundreds
above. But this 30 hundreds also contains all the hundreds
which come from the 3d and 4th parts of the cube of 16. If it
were not so, the 30 hundreds, divided by three times the square
of the tens, would give the unit figure exactly

Forming a divisor of three times the square of the tens, we find
the quotient to be ten , but this we know to be too large. Placing
9 in the root and cubing 19, we find the result to be 6859. Then
trying 8 we find the cube of 18 still too large ; but when we take
6 we find the exact number. Hence the cube root of 4096 is 16.

301. Hence, to find the cube root of a number,

RULE. I. Separate the given number into periods of three
figures each, by placing a dot over the place of units, a second
over the place of thousands, and so on over each third figure
to the left ; the left hand period will often contain less than
three places of figures.

IT. Note the greatest perfect cube in the first period, and
set its root on the right, after the manner of a quotient in di-
vision. Subtract the cube of this n umber from the first period,
and to the remainder bring down the first figure of the next
period for a dividend.

III. Take three times the square of the root just found for
a trial divisor, and see how often it is contained in the divi-
dend, and place the quotient for a second figure of the root.
Then cube the figures of the root thus found, and if their
cube be greater than the first two periods of the given num-
ber, diminish the last figure, but if it be less, subtract it
from the first two periods, and to the remainder bringdown
the first figure of the next period for a new dividend.

IY. Take three times the square of the whole root for a
second trial divisor, and find a third figure of the root.
Cube the whole root thus found and subtract the result from
the first three periods of the given number when it is less
than that number, but if it is greater, diminish the figure
of the root / proceed in a similar way for all the periods.



288 CUBE ROOT.

EXAMPLES.

1. What is the cube root of 99252841 ?

99 252 847(463
4 3 =64

4? x 3=48)352 dividend.
First two periods 99 252

(46)*=46x 46x46= 97 336

3 x (46) 2 =634S ) 19T68 2d dividend.
The first three periods - 99 252 847

(463) 3 =99 252 847
Find the cube roots of the following numbers :



1. Of 389017?

2. Of 5735339?

3. Of 32461759?



4. Of 84604519?

5. Of 259694072?

6. Of 48228544?



302. To extract the cube root of a decimal fraction.

Annex ciphers to the decimal, if necessary, so that it
shall consist of 3, 6, 9, &c., places. Then put the first point
over the place of thousandths, the second over the place of
millionths, and so on over every third place to the right ;
after which extract the root as in whole numbers.

NOTES. 1. There will be as many decimal places in the root
as there are periods in the given number.

2. The same rule applies when the given number is composed
of a whole number and a decimal.

3. If in extracting the root of a number there is a remainder
after all the periods have been brought down, periods of ciphers
may be annexed by considering them as decimals.

EXAMPLES.

Find the cube roots of the following numbers.:



1. Of .157464?
2. Of .870983875 ?
3. Of 12.977875?


4. Of .751089429?
f>. Of .353393243 ?
6. Of 3.408862625?



301. What is the rule for extracting the cube root ?

303. How do you extract the cube root of a decimal fraction ? How
many decimal places will there be in the root ? Will the same rulft
apply when there is a whole number and a decimal ? If in extracting
the root of any number you find i decimal, how do you proceed ?



APPLICATIONS. 289

303. To extract the cube root of a common fraction.

I. Reduce compound fractions to simple ones, mixed num-
bers to improper fractions, and then reduce the fraction to
its lowest terms.

II. Extract the cube root of the numerator and denomi-
nator separately, if they have exact roots ; but if either of
them has not an exact root, reduce the fraction to a decimal
and extract the root as in the last case,

EXAMPLES.

Find the cube roots of the following fractions :

1. Offf|? 4. Of?

2. Of31J&? 5. Off?
3- Of T 3^? 6. Of |?

APPLICATIONS.

1. What must be the length, depth, and breadth of a box,
when these dimensions are all equal and the box contains
4913 cubic feet ?

2. The solidity of a cubical block is 21952 cubic yards :
what is the length of each side ? What is the area of the
surface ?

3. A cellar is 25 feet long 20 feet wide, and 8| feet deep :
what will be the dimensions of another cellar of equal capacity
in the form of a cube ?

4. What will be the length of one side of a cubical granary
that shall contain 2500 bushels of grain ?

5. How many small cubes of 2 inches on a side can be
sawed out of a cube 2 feet on a side, if nothing is lost in
sawing ?

6. What will be the side of a cube that shall be equal to
the contents of a stick of timber containing 1728 cubic feet?

7. A stick of timber is 54 feet long and 2 feet square :
what would be its dimensions if it had the form of a cube ?

NOTES. 1. Bodies are said to be similar when their like parts
are proportional.

2. It is found that the contents of similar bodies are to each
other as the cubes of their like dimensions.

303. How do you extract the cube root of a vulgar fraction ?
19



290 ARITHMETICAL PROGRESSION,

3, All bodies named in the examples are supposed to be simi


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