244. When either or both of the radicals are connected with other quantities by the sign or, each term of the multiplicand must be multiplied by each term of the multiplier (Art. 68).

9. Multiply a +2b by ab.

11. Multiply a+c by ✔a-cb. с

Ans. a- c2b2.

12. Multiply a +√b+√c by √a+√√√c. Ans. abc + 2 √ ab.

DIVISION OF RADICALS.

245. Division of radicals depends upon the principle

that

The quotient of like roots of two quantities is equal to the same root of their quotient.

That is, let a and b represent two quantities; then,

Therefore, if cdab and db be two radicals with coefficients and a common index; then

Reduce the radical parts, if necessary, to those having a common index; then divide the coefficient of the dividend by the coefficient of the divisor, and the radical part of the dividend by that of the divisor, and prefix the first quotient to the last, written under the common index.

Ans. 2.

10. Divide 20 + √12 by ✔5+√3.

11. Divide a+b− c + 2 √ ab by √ã+✔ō−√c.

246. Involution of radicals depends upon the same general principles as involution of rational quantities. That is, let a represent any quantity; then, we have

Therefore, if ba be a radical with a coefficient, then (Art. 200),

Raise the coefficient, if any, to the required power, and write the radical part, raised to the same power, under the given sign.

If a quantity have fractional exponents, the involution may be performed by multiplying them by the exponent of the required power (Art. 203).

If the radical has quantities connected by + or perform the involution by multiplication of the several

3. Raise a b2 to the second power.
4. Raise3 to the third power.
5. Raise

a2 to the fourth power.

ROOTS OF RADICALS.

247. Evolution of radicals depends upon the same general principles as evolution of rational quantities.

That is, let a represent any quantity; then, by Art. 229, we have for the mth root of the nth root of a,

Let it now be required to find a root of a radical with a coefficient, as the cube root of 8 √ a3, or the square root of 205a; then, by Art. 212, we have

√205 a = √ 4 x 5 √5a;

and √√√5 √5a = 2 √ √ 125 a = 2 √125 a.

Hence the

RULE.

Extract the required root of the coefficient, if any, when a complete power of the required degree, otherwise introduce it under the radical sign.

Extract the required root of the quantity under the radical sign, when a complete power of the required degree, otherwise multiply the index of the radical by the index of the required

root.

When only a factor of the coefficient is a complete power of the same degree as the root required, take the root of that factor, and introduce the other factor under the sign.

When a quantity is affected by a fractional exponent, its evolution may be performed by dividing this exponent by the index of the required root (Art. 223).