The same relation holds true whether the exponents are posi-
tive or negative, integral or fractional ; that is, universally,

p r pr

a^-^a^^a^ \
For the quotient must be a quantity which, multiplied by
the divisor, shall produce the dividend ; and, according to Art.
225, the exponent of any letter in a product is in all cases
equal to the algebraic sum of the exponents of this letter in
the multiplicand and multiplier. Hence this relation must
hold true universally in division.
EXAMPL


ES.
1. Divide {aby by (oJ)*.
Ans. {ahf.
2. Divide a* by a*.
3. Divide 4 Voi by 2 Va*.
Ans. 2 Vab.
4. Divide 9m\a^b)^ by 8m(a-

5. Divide a*6* by a*6^.
-')*.
Ans. Sm{a^b)^.
Ans. ah'^.
6. Divide 4* by 2\




Ans. —=,
V2



Division of Polynomial Radicals.
229. By combining the preceding rules with that for the di-
vision of polynomials, Art 80, we may divide one radical ex-
pression by another containing any number of terms.
Ex. 1. Let it be required to divide (i^—cfi—cfi+a^ by a*— 1.
CL^—ci^^ofl+d^ I g^— 1

d^—ct^ a^—a^f Ans.
Ex.2. Divide 8g-6 by 2g*-6*

Ans. 4g^+2a^Z>^+ J*.
164 ALOBBBA.
Ex- 8. Divide a-41a^-120 by a^+4a^+5.
-4n5.a^ --4a^+lla^-.24

Ex. 4. Divide a*+64J^ by a*+46*

Ans. a*-4a*6*+16&*

Ex. 5. Divide x^-^a^+x^y-^y^ by x^-^y^.
^, ^rw. x-\-y.
To involve a Badtcal Quantity to any power.
280. Let it be required to raise a^ to the nth power.
i i i t 1 2
The square of a** is a^xa^=a^ "*=a*".
lilt 1,1,1 8

L ' ML I «l



The cube of a** is a^xa^xa'^=a'^ ^ ^=a^.
