Multiplication of Radicals
273. Radical quantities may be multiplied, like other quantities, by writing the factors one after another, either with or without the sign of multiplication between them. (Art. 91.)
Thus the product of √a into √b, is √a.√b.
The product of n1/3 into y1/2 is h1/3y1/2.
But it is often expedient to bring the factors under the same radical sign. This may be done, if they are first reduced to a common index.
Thus n√x.n√y = n√xy. For the root of the product of several factors is equal to the product of their roots. (Art 254.) Hence,
274. Quantities under the same radical sign or index, may be multiplied together like rational quantities, the product being placed under the common radical sign or index.
Multiply √x into 3√y, that is, x1/2 into y1/3.
The quantities reduced to the same index, (Art. 264.) are (x3)1/6, and (y2)1/6 and their product is, (x3y2)1/6 = 6√.
|Mult.||√a + m||a3/2||a1/m|
In this manner the product of radical quantities often becomes rational. Thus the product of √2 into √18 = √36 = 6.
275. Roots of the same letter or quantity may be multiplied, by adding their fractional exponents.
The exponents, like all other fractions, must be reduced to a common denominator, before they can be united in one term. (Art. 145.)
Thus a1/2.a1/3 = a1/2+1/3 = a2/6+3/6 = a5/6.
The values of the roots are not altered, by reducing their indices to a common denominator. (Art. 250.)
And in all instances of this nature, the common denominator of the indices denotes a certain root; and the sum of the numerators, shows how often this is to be repeated as a factor to produce the required product.
276. From the last example it will be seen, that powers and roots may be multiplied by a common rule. This is one of the many advantages derived from the notation by fractional indices. Any quantities whatever may be reduced to the form of radicals, (Art. 263,) and may then be subjected to the same modes of operation.
Thus y3.y1/6 = y3+1/6 = y19/6.
And x.x1/n = x1+1/n = x(n+1)/n.
Thus a3.a1/3.a2/3 = a12/3 = a4.
And a3/5.a2/5 = a5/5 = a.
277. When radical quantities which are reduced to the same index, have rational coefficients, the rational parts may be multiplied together, and their product prefixed to the product of the rqadical parts.
1. Multiply a√b into c√d.
The product of the rational parts is ac.
The product of the radical parts is √bd.
And the whole product is ac√bd.
By Art. 99,.a.√b into c.√d is a.√b.c.√d or by changing the order of the factors,
a.c.√b.√d = ac.√bd = ac√bd.
But in cases of this nature we may save the trouble of reducing to a common index, by multiplying as in Art. 273.
|Mult.||a(b + x)1/2||a√x||x3√3|
|Into||y(b - x)1/2||b√x||y3√9|
|Prod.||ay(b2 - x2)1/2||abx||3xy|
278. If the rational quantities, instead of being coefficients to the radical quantities, are connected with them by the signs + and -, each term in the multiplier must be multiplied into each in the multiplicand, as in Art. 97.
1.Multiply √a into 3√b. Ans. 6√ .
2.Multiply 5√5 into 3√8. Ans. 30√10.
3. Multiply 2√3 into 33√4. Ans. 66√432.
4. Multiply √d into 3√ab. Ans. 6√