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We have seen above, that when the root is to consist of several figures, the same course is to be pursued as when it consists of only two.

Operation. 3 a' = 270000 34,965,783 (300 + 20 + 7 = 327. 3ab = 18000 27 . . . . . . b” = 400 79,65 (2700 1st divisor. 288400 57 68 20 = 5

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13. What is the third root of 1990865512
14. What is the third root of 513,345,176,343 *
15 What is the third root of 217,125,148,004,864?

XXXII. The third power of a fraction is found by raising both numerator and denominator to the third power. Thus the third power of ; is 3 × 3 × 3 = #.

Hence the third root of a fraction is found by finding the

third root of both numerator and denominator. The third of

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It was remarked with regard to the second root that, when a whole number has not an exact root in whole numbers, its root cannot be exactly found, for no fractional quantity multiplied by itself can produce a whole number. The same is true with regard to all roots, and for the same reason.

Hence the third root of * cannot be found exactly because the numerator has no exact third root. The root of the denominator is 2, that of the numerator is between 2 and 3, nearest to 3. The approximate root is ; or 14.

6. What is the third root of #2

In this, neither the numerator nor the denominator is a perfect third power ; but the denominator may be rendered a perfect third power, without altering the value of the fraction, by multiplying both terms of the fraction by 49, the second power of the denominator.

3 × 49 147 7 × 49 343

The root of this is between # and #, nearest to the former.

It is evident that the denominator of any fraction may be rendered a perfect third power, by multiplying both its terms by the second power of the denominator. The third root of a whole number which is not a perfect third power, may be approximated by converting the number into a fraction, whose denominator is a perfect third power.

What is the third root of 5 °

We may find this root exact within less than or of a unit, by converting it into a fraction, whose denominator is the third power of 12.

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The most convenient numbers to multiply by, are the third powers of 10, 100, 1000, &c. in which case, the fractional part of the root will be expressed in decimals, in the same manner as was shown for the second root. The multiplication may be performed at each step of the work. For each decimal to be obtained in the root, three zeros must be annexed to the number, because the third power of 10 is 1000, that of 100, 1000000, &c.

7. The third root of 5 will be found by this method as fol

lows.

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The 3d root of 5 is 1.709, within less than rior of a unit. We might approximate much nearer if necessary. The other method explained in the last article may be used if preferred.

8. What is the third root of 17; 2

The fractional part of this number must first be changed to a decimal. 173 = 17.75 = Ho - 17.750.

Hence it appears, that to prepare a number containing degimals, it is necessary that for every decimal place in the root, there should be three decimal places in the power. Therefore we must begin at the place of units, and separate the number both to the right and left into periods of three figures each. If these do not come out even in the decimals, they must be supplied by annexing zeros to the right.

9. What is the approximate third root of 25732.75 ° 10. What is the approximate third root of 23.1702 * 11. What is the approximate third root of 12; ? 12. What is the approximate third root of 144 13. What is the approximate third root of ## * 14. What is the approximate third root of or ?

XXXIII. Questions producing Pure Equations of the Third

egree.

1. A man wishes to make a cellar, that shall contain 31 104 cubic feet; and in such a form, that the breadth shall be twice the depth, and the length 1; the breadth. What must be the length, breadth, and depth :

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and the length =#. The whole content will be a X 2 r x * =suo 193 = 31104 3

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2. There are two men whose ages are to each other as 5 to 4, and the sum of the third powers of their ages is 137781. What are their ages?

Let a = the age of the elder

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3. A man wishes to make a cubical cistern that shall contain 100 gallens. What must be the length of one of its sides 2

4. A bushel is 2150} cubic inches. What must be the size of a cubical box to hold 1 bushei ?

5. What must be the size of a cubical box to hold 2 bushels?

6. What must be the size of a cubical box to hold 8 bushels?

7. Find two numbers, such that the second power of the greater multiplied by the less may be equal to 448; and the second power of the less multiplied by the greater, may be 302 P

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