Addition and Subtraction of Radicals

268. Radical quantities may be added like rational quantities, by uniting them one after another with their signs. (Ait. 68.)
Thus the sum of √a and √b, is √a + √b.
And the sum of a1/2 - h1/3 and x1/4 - y1/n, is a1/2 - h1/3 + x1/4 - y1/n.

But in many cases, several terms may be reduced to one as in Arts. 71 and 73.
The sum of 2√a and 3√a is 2√a + 3√a = 5√a.
For it is evident that twice the root of a, and three times the root of a, are five times the root of a. Hence,

269. When the quantities to be added have the same radical part, under the same radical sign or index; add the rational parts, and to the sum annex the radical parts.

If, no rational quantity is prefixed to the radical sign, 1 is always to be understood. (Art. 240.)

To 2na 3(x + h)1/7 a√b - h
Add nay 4(x + h)1/7 y<√b - h
Sum 3nay 7(x + h)1/7 (a + y)√b - h

270. If the radical parts are originally different, they may sometimes be made alike, by the reductions in the preceding articles.

1. Add √8 to √50. Here the radical parts are not the same. But by the reduction in Art. 266, √8 = 2√2, and √50 = 5√2. The sum then is 7√2.

2. Add √16b to √4b. Ans. 4√b + 2√b = 6√b.

3. Add √a2x to √b4x. Ans. a√x + b2x = (a + b2).√x.

4. Add √18a to 3√2a.

271. But if the radical parts, after reduction, are different or have different exponents, they cannot be united in the same term; and must he added by writing them one after the other.
The sum of 3√b and 2√a, is 3√b + 2√a.

It is manifest that three times the root of b, and twice the root of a, are neither five times the root of b, nor five times the root of a, unless b and a are equal.
The sum of √a and 3a, is √a + 3a.

The square root of a, and the cube root of a, are neither twice the square root, nor twice the cube root of a.

272. Subtraction of radical quantities is to be performed in the same manner as addition, except that the signs in the subtrahend are to be changed according to Art. 81.

From ay 3h1/3 -a-1/n
Sub. 3√ay -5h1/3 -2a-1/n
Diff. -2√ay 8h1/3 a-1/n

From √50, subtract √8. Ans. 5√2 - 2√2 = 3√2. (Art. 270.)
From 3b4y subtract 3by4. Ans. (b - y).3x.
From nx, subtract 5x.

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