Multiplication.

111. For the multiplication of radicals, we have the following

RULE.

I. Multiply the quantities under the radical signs together, and place the common radical over the product.

II. If the radicals have coefficients, we multiply them together, and place the product before the common radical.

This is the principle of Art. 104, taken in the inverse order.

EXAMPLES.

1. What is the product of 3√/5ab and 4√/20a ?

Ans. 120a 5.

2. What is the product of 2a√be and 3a√bc?

Ans. 6a2bc.

3. What is the product of 2a ya+b2 and -3a √a2 + b2? Ans. -6a2(a+b2).

QUEST.-111. How do you multiply quantities which are under radical signs? When the radicals have coefficients, how do you multiply them ?

4. What is the product of 32 and 28.

Ans. 24.

5. What is the product of √3a2 and√cb?

Ans. abc 15.

6. What is the product of 2x+√b and 2x-√b?

7. What is the product of

Ans. 4x2-b.

√a+2√b and √a-2√b?

Ans. Va2-4b.

8. What is the product of 3a√27a3 by √2a?

Division.

Ans. 9a3√6.

112. To divide one radical by another, we have the following

RULE.

I. Divide one of the quantities under the radical sign by the other, and place the common radical over the quotient.

II. If the radicals have coefficients, divide the coefficient of the dividend by the coefficient of the divisor, and place the quotient before the common radical.

QUEST.-112. How do you divide quantities which are under the radical sign? When the radicals have coefficients, how do you divide them?

expressions are equal to the same quantity; hence

the expressions themselves must be equal.

9. Divide 6a1062 by 35.

10. Divide 486 √15 by 262 VS.

11. Divide 8a2b4c37d3 by 2a√28d.

Ans. a3b2.

Ans. 2aby/2.

Ans. 360b2.

Ans. 2ab+c3d.

12. Divide

96a4c3 √9865 by 48abc √26.

Ans. 14a3b2c2.

13. Divide 27a5b6 √21a3 by √Ta..

Ans. 27ab6 √3.

14. Divide 18a8668 by 6ab ya.

Ans. 6a8b5 √2.

To Extract the Square Root of a Polynomial.

113. Before explaining the rule for the extraction of the square root of a polynomial, let us first examine the squares of several polynomials: we have

(a+b)2=a2+2ab+b2,

(a+b+c)2=a2+2ab+b2+2(a+b)c+c2,

(a+b+c+d)2=a2+2ab+b2+2(a+b)c+c2
+2(a+b+c)d+d2.

The law by which these squares are formed can be enunciated thus:

The square of any polynomial contains the square of the first term, plus twice the product of the first term by the second, plus the square of the second; plus twice the first two terms multiplied by the third, plus the square of the third; plus twice the first three terms multiplied by the fourth, plus the square of the fourth; and so on.

QUEST.-113. What is the square of a binomial equal to? What is the square of a trinomial equal to? What is the square of any polynomial equal to ?

114. Hence, to extract the square root of a polynomial we have the following

RULE.

I. Arrange the polynomial with reference to one of its letters and extract the square root of the first term: this will give the first term of the root.

II. Divide the second term of the polynomial by double the first term of the root, and the quotient will be the second term of the root.

III. Then form the square of the two terms of the root found, and subtract it from the first polynomial, and then divide the first term of the remainder by double the first term of the root, and the quotient will be the third term.

IV. Form the double products of the first and second terms, by the third, plus the square of the third; then subtract all these products from the last remainder, and divide the first term of the result by double the first term of the root, and the quotient will be the fourth term. Then proceed in the same manner to find the other terms.

EXAMPLES.

1. Extract the square root of the polynomial

49a2b2-24ab3+25a4-30a3b+16b1.

First arrange it with reference to the letter a.