# 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 a^{1/2} - h^{1/3} and x^{1/4} - y^{1/n}, is a^{1/2} - h^{1/3} + x^{1/4} - y^{1/n}.

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 | 2^{n}√a |
3(x + h)^{1/7} |
a√b - h |

Add | ^{n}√ay |
4(x + h)^{1/7} |
y<√b - h |

Sum | 3^{n}√ay |
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 √a^{2}x to √b^{4}x. Ans. a√x + b^{2}√x = (a + b^{2}).√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 ^{3}√a, is √a + ^{3}√a.

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 | 3h^{1/3} |
-a^{-1/n} |

Sub. | 3√ay | -5h^{1/3} |
-2a^{-1/n} |

Diff. | -2√ay | 8h^{1/3} |
a^{-1/n} |

From √50, subtract √8. Ans. 5√2 - 2√2 = 3√2. (Art. 270.)

From ^{3}√b^{4}y subtract ^{3}√by^{4}. Ans. (b - y).^{3}√x.

From ^{n}√x, subtract ^{5}√x.