NOTE. - Dividing the index of the root is the same as multiplying the fractional exponent. Thus the square of va is a. For (a3)2 — a3, or ya.

the cube root of the radical part the cube root of the rational part. The cube root of the radical part must be a quantity which, taken three times as a factor, will produce

a3x3; i. e. ax.

In this case we have used the fractional exponent, and found the fourth root by dividing the exponent of the

given quantity by the index of the required root, according to

Art. 143. Hence,

RULE.

I. Evolve the radical as if it were rational, and, placing it under its proper radical sign, prefix the required root of its coefficient.

II. A radical can be evolved by dividing its fractional exponent by the index of the required root.

NOTE.

Multiplying the index of the root is the same as dividing the fractional exponent. Thus, the square root of a is va. For (a) at, or Va.

=

3. Find the square root of 5 a 4x.

(5 a ✅4x)1 = (✅ 500 a3 x)* = ✅ 500 a3 x, Ans.

POLYNOMIALS HAVING RADICAL TERMS.

163. It appears from the principles already established, that the laws which apply to calculations with quantities which have exponents, apply equally well whether the exponents are positive or negative, integral or fractional. The following examples, therefore, can be done by rules already given.

1. Add 4a-3√√y and 3 a +2y. Ans. 7a-Ny. 2. Add 3x+135 and 7x-1080.

Ans. 10x35.

3. Add 228-27 and 2No63 +No48.

4. Subtract 15 x

50 a from 13 x

8a.
Ans. 32a-2x.

5. Subtract ax2-4b from ax-166.

8. Multiply xy+ab by 4-✔ab.

Ans. 4xy (4 xy) √ ab — ab.

Ans. 3210.

9. Multiply 7+10 by 6-10.

10. Multiply a +√√bby√a-√√b.

Ans. a b.

~3-77

11. Multiply ✔5–43 by 45 +9.

12. Multiply √ễ +7√3 by 1√≥ −7 √3.

13. Divide ✔ax +✔ay+x+√xy by √a+√x.

OPERATION.

√a + √x) √ a x + √ay+x+√xy√x+√ÿ

14. Divide acad√bc + √bd by √c-✔d. Ans. ab.

15. Divide ax+x+a3y+b1y by x+y.

16. Divide xy by x-y. Ans. y.

17. Divide 4 xy +4√ab-xy√ab-ab by 4-ab.

SECTION XVIII.

PURE EQUATIONS

WHICH REQUIRE IN THEIR REDUCTION EITHER INVOLUTION OR EVOLUTION.

164. A PURE EQUATION is one that contains but one power of the unknown quantity; as,

√x+ac=b, 4x2 + 3 = 7, or 14 x" = a b.

4x2+3=

165. A PURE QUADRATIC EQUATION is one that contains only the second power of the unknown quantity; as,

166. RADICAL EQUATIONS, i. e. equations containing the unknown quantity under the radical sign, require Involution in their reduction.