The values will both be positive in this, and both answer th conditions.

The values are 5 and 1 as before, but now both are positive and both answer the conditions of the question.

3. There are two numbers whose sum is a, and the sum of whose second powers is b. It is required to find the numbers

Examine the various cases which arise from giving differen values to a and b. Also how the negative value is to be inter preted. Do the same with the following examples.

4. There are two numbers whose difference is a, and the sum of whose second powers is b. Required the numbers.

5. There are two numbers whose difference is a, and the difference of whose third powers is b. Required the numbers.

6. A man bought a number of sheep for a number a of dol lars; and on counting them he found that if there had been number b more of them, the price of each would have been less by a sum c. How many did he buy?

7. A grazier bought as many sheep as cost him a sum a, ou of which he reserved a number b, and sold the remainder for a sum c, gaining a sum d per head by them. How many sheep did he buy, and what was the price of each?

8. A merchant sold a quantity of brandy for a sum a, and gained as much per cent. as the brandy cost him. What was the price of the brandy?

XXXVI. Of Powers and Roots in General.

Some explanation of powers both of numeral and literal quantities was given Art. X. The method of finding the roots of the second and third powers, that is, of finding the second and third roots of numeral quantities, has also been explained and their application to the solution of equations. But it is

en

ter

frequently necessary to find the roots of other powers, as well as of the second and third, and of literal, as well as of numeral quantities. Preparatory to this, it is necessary to attend a little more particularly to the formation of powers.

The second power of a is a X a = a2.

The fifth power of a is a X a X a × a × a = a3.

If a quantity as a is multiplied into itself until it enters m times as a factor, it is said to be raised to the mth power, and is expressed am. This is done by m-1 multiplications; for one multiplication as a X a produces a2 the second power, two multiplications produce the third power, &c.

We have seen above Art. X. that when the quantities to be multiplied are alike, the multiplication is performed by adding the exponents. By this principle it is easy to find any power of a quantity which is already a power. Thus

The second power of a3 is a3 X a3 = a13 = ao.

am+m+m

3m

= a3m

The third power of a' is a2 X a × a2 = a2+2+3 = a®. The second power of am is am × am = am+m = a2m. The third of am is am X am X am = a' power The mth power of a2 is a X a2 × a2 × a3 × =ɑ2+2+a+a+. , until a2 is taken m times as a factor, that is, until the exponent 2 has been taken m times. Hence it is expressed am.

.....

The nth power of am is am × am × am = am+m+m+. until m is taken n times, and the power is expressed am.

...

N. B. The dots..... in the two last examples are used to express the continuation of the multiplication or addition, because it cannot come to an end until m in the first case, and n in the second, receive a determinate value.

In looking over the above examples we observe ;

1st. That the second power of a3 is the same as the third power of a2, and so of all others.

2. That in finding a power of a letter the exponent is added until it is taken as many times as there are units in the exponent of the required power. Hence any quantity may be raised to any power by multiplying its exponent by the exponent of the power to which it is to be raised,

The 5th power of a3 is a3×3 = a1.

The 3d power of a' is a1×3 = aa1, &c.

The power of a product is the same as the product of that power of all its factors.

The 2d power of 3 a b is 3 ab × 3 ab = 9 a2 b3.

The 3d power of 2 a2 b3 is 2 a2 b3 × 2 a2 b3 × 2 a2 b3 = 8 ao bo.

Hence, when a quantity consists of several letters, it may be raised to any power by multiplying the exponents of each letter by the exponent of the power required; and if the quantity has a numeral coefficient, that must be raised to the power required.

The powers of a fraction are found by raising both numerator and denominator to the power required; for that is equivalent to the continued multiplication of the fraction by itself.

1 What is the 5th power of 3 a2 b3 m?

Powers of compound quantities are found like those of simple quantities, by the continued multiplication of the quantity into itself. The second power is found by multiplying the quantity once by itself. The third power is found by two multiplications, &c,

The powers of compound quantities are expressed by enclosing the quantities in a parenthesis, or by drawing a vinculum over them, and giving them the exponent of the power. The third power of a+ 2 b c is expressed (a + 2 b —c)3 ; or

-

The powers are found by multiplication as follows:

a2 + 4 ab + 4 b2 — 2 a c— 4 b c + c2 = (a + 2 b — c)2

a +26 C

a3+4a3b+4 a b2-2 a2c-4abc + ac2

2ab+8 ab2+8 b3 — 4 abc-8b°c + 2 b c2

-ac-4abc-4b2c+2 ac2+4bc2-c2

a3 + 6 a2 b + 12 a b2 + 8 b3 — 3 a' c— 12 abc-12 b2 c +3 ac2+6b c2 — c3 = (a + 2 b —c)3.

If the third power be multiplied by a +2b-c, it will produce the fourth power.

3. What is the second power of 3 c + 2 d?

4. What is the third power of 4 a-bc?

6. What is the fourth power of 2 a2 c— c2?

In practice it is generally more convenient to express the powers of compound quantities, than actually to find them by multiplication. And operations may frequently be more easily performed on them when they are only expressed.

(a + b)3 × (a + b)2 = (a + b)3+2 = (a + b)3

(3 a 5 c) (3 a 5 c)3 (3 a-5 c).

That is, when one power of a compound quantity is to be multiplied by any power of the same quantity, it may be expressed by adding the exponents, in the same manner as simple quantities.

The 2d power of (a + b)3 is (a + b)3 × (a + b)3

==

· (a+b)3±3 =

(a + b)3×2 = (a + b)°.

The 3d power of (2 a — d)* is

4 X 3

(2 a — d)1+4+4 = (2 a — d) 1×3 — (2 a — d)12.

That is, any quantity, which is already a power of a compound quantity, may be raised to any power by multiplying its exponent by the exponent of the power to which it is to be raised.

7. Express the 2d power of (3b-c)*.

8. Express the 3d power of (a — c + 2 d)3.

9. Express the 7th power of (2 a2 — 4 c3)3.

Division may also be performed by subtracting the exponents as in simple quantities.

(3 a—b)3 divided by (3 a—b)3 is

10. Divide (7 m + 2 c)7 by (7 m + 2 c)3.

If (a+b) is to be multiplied by any quantity c, it may be expressed thus: c (a+b)2. But in order to perform the operation, the 2d power of a + b must first be found.

c (a + b)2 = c (a2 + 2 a b + b2) = a°c + 2 a b c + b2 c

If the operation were performed previously, a very erroneous result would be obtained; for c (a+b) is very different from (ac+bc). The value of the latter expression is a2 c2 + 2 ab c2 + b2 c2.

11. What is the value of 2 (a + 3b)' developed as above? 12. What is the value of 3 b c (2 a —

- c)2?

13. What is the value of (a + 3 c2) (3 a — 2b)2 ?

14. What is the value of (2 a—b)2 (a2 + bc)2?