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 ^{3}√a divided by √b, is ^{3}√a/√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 ^{n}√a:^{n}√b = ^{n}√a/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, ^{n}√ab:^{n}√b = ^{n}√a. For the product of this quotient into the divisor is equal to the dividend, that is,
^{n}√a.^{n}√b = ^{n}√ab. 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 (x^{3}y^{2})^{1/6} by y^{1/3}.
These reduced to the same index are (x^{3}y^{2})^{1/6} and (y^{2})^{1/6}:
And the quotient is (x^{3})^{1/6} = x^{3/6} = x^{2}.
Divide | √6a^{3}x | (a^{3} + ax)^{1/9} | (a^{2}y^{2})^{1/4} |
By | √3x | a^{1/9} | (ay)^{1/4} |
Quot. | √2a^{3} | (a^{2} + 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 A^{1/2}:a^{1/6} = a^{1/2-1/6} = a^{3/6-1/6} = a^{2/6} = a^{1/3}.
Divide | (3a)^{11/12} | a^{(m+n)/mn} | (r^{2}y^{3})^{1/7} |
a^{2/3} | a^{1/m} | (r^{2}y^{3})^{3/7} | |
Quot. | (3a)^{1/4} | a^{1/n} | (r^{2}y^{3})^{-2/7} |
Powers and roots may be brought promiscuously together, and divided according to the same rule. See Art. 281.
Thus a^{2}:a^{1/3} = a^{2-1/3} = a^{5/3}. For a^{5/3}.a^{1/3} = a^{6/3} = a^{2}
So y^{n}:y^{1/m} = y^{n-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(a^{3}x^{2})^{1/n} | b√xy |
By | 6√a | y(AX)^{1/n} | √y |
Quot. | 4x√y | b(a^{2}x)^{1/n} | b√x |
Divide ab(x^{2}b)^{1/4} by a (x)^{1/2}.
These reduced to the same index are ab(x^{2}b)^{1/4} and a(x^{2})^{1/4}.
The quotient then is b(b)^{1/4} = (b^{5})^{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(x^{2}b)^{1/4}]/a(x)^{1/2}.
1. Divide 2^{3}√bc by 3√ac.
Ans. (2/3)^{6}√b^{2}/a^{3}c.
2. Divide 10^{3}√108 by 5^{3}√4.
Ans. 2^{3}√27 = 6.
3. Divide 10√27 by 2√3.
Ans. 15.
4. Divide 8√108 by 2√6.
Ans. (ab)^{1/3}.