Division of Radicals

279. The division of radical quantities may be expressed, by writing the divisor under the dividend, in the form of a fraction.

Thus the quotient of 3a divided by √b, is 3a/√b.

In this instance, the radical sign or index is separately applied to the numerator and the denominator. But if the divisor and dividend are reduced to the same index or radical sign, this may be applied to the whole quotient.

Thus na:nb = na/b. For the root of a fraction is equal to the root of the numerator divided by the root of the denominator. (Art. 256.)

Again, nab:nb = na. For the product of this quotient into the divisor is equal to the dividend, that is,
na.nb = nab. Hence,

280. Quantities under the same radical sign or index may be divided like rational quantities, the quotient being placed under the common radical sign or index.

Divide (x3y2)1/6 by y1/3.
These reduced to the same index are (x3y2)1/6 and (y2)1/6:
And the quotient is (x3)1/6 = x3/6 = x2.

Divide 6a3x (a3 + ax)1/9 (a2y2)1/4
By 3x a1/9 (ay)1/4
Quot. 2a3 (a2 + x)1/9 (ay)1/4

281. A root is divided by another root of the same letter or quantity, by subtracting the index of the divisor from that of the divident.
Thus A1/2:a1/6 = a1/2-1/6 = a3/6-1/6 = a2/6 = a1/3.

In the same manner, it may be shown that o1/m:a1/n = a1/m-1/n.
Divide (3a)11/12 a(m+n)/mn (r2y3)1/7
a2/3 a1/m (r2y3)3/7
Quot. (3a)1/4 a1/n (r2y3)-2/7

Powers and roots may be brought promiscuously together, and divided according to the same rule. See Art. 281.
Thus a2:a1/3 = a2-1/3 = a5/3. For a5/3.a1/3 = a6/3 = a2
So yn:y1/m = yn-1/m.

282. When radical quantities which are reduced to the same index have rational coefficients, the rational parts may be divided separately, and their quotient prefixed to the quotient of the radical parts.
Thus ac√bd:a√b = c√d. For this quotient multiplied into the divisor is equal to the dividend.

Divide 24x√ay by(a3x2)1/n b√xy
By 6√a y(AX)1/n y
Quot. 4x√y b(a2x)1/n b√x

Divide ab(x2b)1/4 by a (x)1/2.

These reduced to the same index are ab(x2b)1/4 and a(x2)1/4.
The quotient then is b(b)1/4 = (b5)1/4 (Art 267.)

To save the trouble of reducing to a common index, the division may be expressed in the form of a fraction.
The quotient will then be [ab(x2b)1/4]/a(x)1/2.

1. Divide 23bc by 3√ac.
Ans. (2/3)6b2/a3c.

2. Divide 103108 by 534.
Ans. 2327 = 6.

3. Divide 10√27 by 2√3.
Ans. 15.

4. Divide 8√108 by 2√6.

5. Divide (a2b2d3)1/6 by d1/2.
Ans. (ab)1/3.


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