2. Required the square root of 18-2√77.

Here, a=18, b=308, and c=324-308=4; hence,

3. Required the square root of 94 + 42√√5.

Ans. 7 +3√√5.

4. Required the square root of 28 + 10√3.

Ans. 5 + √√3.

OPERATIONS ON IMAGINARY QUANTITIES.

137. An IMAGINARY QUANTITY is an indicated even root of a negative quantity. Thus,

--

4

√ 4, √ 16,

V

a2, are imaginary quantities. The rule deduced for multiplying radicals requires some modification, when applied to imaginary quantities By the rule already deduced, the product of √ 4 by √3 would be equal to 12; whereas, the true product is √12, as will be shown hereafter. Every imaginary quantity of the second degree can, by Principle (1), (Art. 125), be resolved into two factors, one of which is V 1 ; the other may be either rational or irrational. Thus,

·4

√√=4 = 2√√−1, √−3 = √√3× √ −1, √—a2 = a√-i.

The factor, 1, is called the imaginary factor, and the other one is called its coefficient. Thus, in the expression, √x 1, the factor 3 is the coefficient

together, we first reduce them to the form, a√ FT. We can then multiply together the coefficients of the imaginary factor by known rules. It remains to deduce a rule for multiplying the imaginary factor into itself, or what is the same, for raising the imaginary factor to a power whose exponent is equal to the number of factors. The first power of √ 1, is -1; the second power, by the definition of square root, is

1; the

third power, is the product of the first and second √1; the fourth power, powers, or

-

-

1 =

is the square of the second, or +1; the fifth, is the product of the first and fourth, that is, it is the same as the first; the sixth, is the same as the second; the seventh, the same as the third; the eighth, the same as the fourth; the ninth, again, the same as the first; and so on indefinitely, as shown in the table, n being any whole number.

To show the use of this table, let it be required to find the continued product of √4, √ — 3, √ − 2, √−7, and 8. Reducing these expressions to the proper form, and indicating the multiplication, we have,

Changing the order of the factors,

(2 × √3 × √2 × √7 × 2√2) (√ — 1)3.

Hence, the product is equal to, 8/21 x 1 = 8√—21.

Ans. 11.

6. (32) × (3 + √− 2).

From what precedes, it follows that the only radı. cal parts of any power of an expression of the form, a± b√-1, will be of the form c√1.

Properties of Imaginary Quantities.

138. 1. A quantity of the form, av.

1, cannot

be equal to the sum of a rational quantity and a quan tity of the form, b√ — 1.

For, if so, let us have the equality,

a√-1 = x + b√ = 1.

Squaring both members, we have,

1 =

an equation which is manifestly absurd, for the first member is imaginary, and the second real, and no imaginary quantity can be be equal to a real quantity. Hence, the hypothesis is absurd; and, consequently, the principle enunciated is true.

In the same way, it may be shown that no radical of the second degree can be equal to an entire quan Lity plus a radical of the second degree.

2. If, a+b√√ − 1 = x + y√ — 1, then a = x, and b = y.

For, by transposition, we have,

But from the preceding principle, this equation can only be true when x α = 0, or x = a; making this supposition, and dividing both members of the given equation by 1, we have by: which was to be shown.

=y:

In the same way, it may be shown that, when a + √o = x +√, we shall have, a = x, and b that is, in all equations of this form, the rational and radical parts, in each member, are respectively equal to each other.

3. The product of two factors, of the forms, 2- (a+b√ 1), and x-(a - b√ — 1), is positive for all real values of x.

For, performing the multiplication, we find the pro duct equal to,

x2 2ax + a2 + b2 ;

which can be written under the form,

(x − a)2 + b2.

-

Now, whatever may be the value of x, the part (-a)2 will be positive, since it is a square; b2 is also positive; hence, their sum, or the required product, is also positive which was to be proved.

RADICAL EQUATIONS.

139. RADICAL EQUATIONS are equations containing radical quantities. No fixed rules can be given for solving such equations. The methods of proceeding will be best illustrated by examples.

EXAMPLES.

1. Given √ + 16 = 2 + √√x, to find x. Squaring,

x + 16 = 4 + 4 √√√x + x.

Whence,

x = 9.

√1 − x = n(1 + √1 − x), to find a

1 — 2n + n2 = (n2 + 2n + 1) (1 − x).