plus three times the tens, multiplied by the square of the units, plus the cube of the units.

1. What is the cube root of 91125?

3. What is the cube root of 2985984 ? 4. What is the cube root of 75686967? 5. What is the cube root of 644972544 ? 6. What is the cube root of 50243409? 7. What is the cube root of 12862247607? 8. What is the cube root of 163039787847? 9. What is the cube root of 50023150823736? 10. What is the cube root of 64240300125 ? 11. What is the cube root of 94996712418949125? 12. What is the cube root of 94997087172244118016?

13. What is the cube root of 163.04?

14. What is the cube root of 3.46?

15. What is the cube root of 3.375?

305. To extract the cube root of a common fraction,

RULE.

Find the cube root of the numerator and the denominator, if it can be done without a remainder; if not, reduce the fraction to a decimal, and then find the nearest cube root.

EXAMPLES.

16. What is the cube root of ?? 17. What is the cube root of 1? 18. What is the cube root of 13? 19. What is the cube root of 914? 20. What is the cube root of ? 21. What is the cube root of ¿?

306. To find two mean proportionals between two given numbers,—

RULE. Divide the greater number by the less, and extract the cube root of the quotient. The less number multiplied by this root will be the least mean proportional. The larger number divided by this root will be the greatest mean proportional.

22. What are the two mean proportionals between 6 and 162?

162627. 273. 3 x 6 18, the least mean proportional; 1623 54, the greatest.

EXAMPLES.

23. What are the two mean proportionals between 11 and 704?

24. What are the two mean proportionals between 12 and 2592 ?

25. What are the two mean proportionals between 8 and 1000 ?

What is the rule for finding two mean proportionals between two given numbers ?

26. What are the two mean proportionals between 9 and 3087 ?

27. What are the two mean proportionals between 25 and 12800?

ROOT OF HIGHER POWERS.

SECTION XXXII.

307. THE root of any power may be found by extracting in succession the roots denoted by the several factors of the index of any higher power. Thus, the fourth root of any number may be found by extracting the square root twice. 34-81. The square root of 81 is 9. The square root of 9 is 3. The sixth root may be found by extracting the square root, and then the cube root; as, 6=2x3; the eighth root, by extracting the square root three times; as, 8=2x2 x2. The ninth root, by extracting the cube root twice; as, 9=3×3. The twelfth, by extracting the square root twice, and then the cube; as, 12=2×2×3. 308. To extract the root of any power,

-

RULE. Separate the given number into periods of as many figures each as there are units in the index of the power, beginning with units.

Find the first figure of the root, and subtract its power from the left hand period, and to the remainder annex the first figure of the next period, for a dividend.

Involve the root to the power next inferior to thứ. denoted by the index, and multiply it by the index, for a divisor.

Find how many times the divisor is contained in

What is the rule for finding the root of higher powers?

the partial dividend, and the result will be the second figure of the root.

Involve the figures of the root already found to the power denoted by the index, and subtract it from the two left hand periods, and to the remainder annex the first figure of the next period, for a dividend, and find the divisor as before. Proceed in this manner till the whole root is found.

1. What is the fifth root of 13542593318343?

2. What is the fourth root of 32015587041 ?

3. What is the seventh root of 2423162679857794647 ?

4. What is the eighth root of 1024997813579847135681 ?

ARITHMETICAL PROGRESSION.

SECTION XXXIII.

309. ARITHMETICAL PROGRESSION is a series of numbers which increase or decrease by a common dif. ference; as, 2, 4, 6, 8, 10, is an increasing series, in which the common difference is 2; 15, 12, 9, 6, 3, is a decreasing series, in which the common difference is 3.

310. The numbers which form the series are called What is arithmetical progression ?

the terms of the progression. The first and last terms are called the extremes, the other terms are called the

means.

311. When any three of the five following terms are known, the remaining two may be found.

312. To find the common difference when the two extremes and the number of terms are known,

RULE. Divide the difference of the extremes by the number of terms, less one, and the quotient will be the common difference.

This rule may be represented by the formula, thus: Let a first term, d=common difference, l=last or squired term, n=the number of terms, s=sum of all he terms.

It is evident that the number of differences will be one less than the number of terms, as there will be one term at each end of the series. It is also evident that the whole difference between the extremes will be the amount of the differences between each term. 1. The extremes are 2 and 56, and the number of ternis 19. Required the common difference.

2. if the extremes be 3 and 95, and the number of terms 24, what is the common difference?

3. If the extremes be 2 and 100, and the number of terms 30, what is the common difference?

4. A man starts on a journey, and travels the first

What is the rule for finding the common difference, when the extremes and number of terms are known?