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

a", &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 a b = 9 a2 b2.

The 3d power of 2 a b3 is 2 a b3 × 2 a2b3 × 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 ?

2 What is the 3d power of 2 ac2

5 b'd3

?

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 bc is expressed (a+2b-c)3; or a+26

3

C.

The powers are found by multiplication as follows:

a+26-c

a+2b-c.

a2+2ab-ac

2ab+4b2bc

—ac―2bc+c2

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

a3 + 4 a2 b + 4 a b2-2 a2c-4abc + ac2

2 a2 b + 8 a b2 + 8 b3 — 4 a b c — 8 b3 c + 2 b c2

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

a3 + 6 a2 b + 12 a b2 + 8 63 — 3 a2c-12 abc-12 b2 c + 3 ac2+6bc2 — c3 = (a + 2 b—c)3.

If the third power be multiplied by a +26-c, it will produce the fourth power.

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

4. What is the third power of 4 a — b c ?

5. What is the fifth power of a-b?

6. What is the fourth power of 2 a3 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—5c)".

That is, when one power of a compound quantity is to be multi plied by any power of the same quantity, it may be expressed by add ing 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 ad) is

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

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 (3 b—c)'.

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

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

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

(3a - b) divided by (3 a—b)3 is

(3 a—b)3 (3 a — b)3

=

10. Divide (7 m + 2 c)' 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). But in order to perform the operation, the 2d power of a + b must first be found.

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

- c)3 ?

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

14. What is the value of (2 a—b)2

(a2 + b c)2?

We have had occasion in the preceding pages to return from the second and third powers to their roots. We have shown how this can be done in numeral quantities; it remains to be shown how it may be effected in literal quantities. It is frequently necessary to find the roots of other powers as well as of the second and third.

The power of a literal quantity, we have just seen, is found by multiplying its exponent by the exponent of the which it is to be raised.

power to

The second power of a3 is a3 x 2 = a; consequently the se

The third power of am is a3m; hence the third root of a3m

In general, the root of a literal quantity may be found by divid ing its erponent by the number expressing the root; that is, by dividing by 2 for the second root, by 3 for the third root, &c. This is the reverse of the method of finding powers.

It was shown above, that any power of a quantity consisting of several factors is the same as the product of the powers of the several factors. From this it follows, that any root of a quantity consisting of several factors is the same as the product of the roots of all the factors.

The third power of a2 b c3 is a b3 c; the third root of ao b3 c9 must therefore be a2 b c3.

Numeral coefficients are factors, and in finding powers they are raised to the power; consequently in finding roots, the root of the coefficient must be taken.

When the exponent of a quantity is divisible by the number expressing the degree of the root, the root can be found exactly; but when it is not, the exponent of the root will be a fraction.

3

The second root of a3 is a The second root of a is a3. The third root of a is at. The nth root of a is a. The nth

m

root of am is añ.

1

The root of a fraction is found by taking the root of its nu merator and of its denominator. This is evident from the method of finding the powers of fractions.

The root of any quantity may be expressed by enclosing it in a parenthesis or drawing a vinculum over it, and writing a fractional exponent over it, expressive of the root. Thus

The 3d root of 8 a3 b is expressed

The root of a compound quantity may be expressed in the

same way.

The 4th root of a2 + 5 ab is expressed

(a2 + 5 ab) ‡

or a2 + 5 a b2.

When a compound quantity has an exponent, its root may be found in the same manner as that of a simple quantity.

The 3d root of (2 b — a)o is (2 b — a) 3 — (2 b —a)2.

=

With regard to the signs of roots it may be observed, that all even roots must have the double sign; for since all even powers are necessarily positive, it is impossible to tell whether the power was derived from a positive or negative root, unless something in the conditions of the question shows it. An even root of a negative quantity is impossible. All odd roots will have the same sign as the power.

15. What is the second root of 9 a2 b® ?

16. What is the third root of 125 a b c ?

17. What is the fifth root of 32 a10 xm r?