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Squaring both members of (1) and (2),

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Adding (3) and (4), and omitting the common fac

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Now, if a b is a perfect square, its root may be represented by c. Substituting in (V) and (8), and extracting the square root of each member of both equations, (axiom 5), we have,

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These values, substituted in (1) and (2), give,

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The square root of the given quantities may be extracted when a?

b is a perfect square, and the roots may be obtained, by substitution, from (9) and (10).

EXAMPLES.

1. Required the square root of 14 +675=14+V180.

Here, a=14, b= 180, and c= V196 – 180 = 4: hence,

V 14 + 6 V5 =

14 + 4

+ 2

14 - 4

2

3+V5.

2. Required the square root of 18 – 2V77.

Here, a = 18, b=308, and c= V324 — 308 = 4; hence,

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3. Required the square root of 94 + 42V5.

Ans. y + 375.

4. Required the square root of 28 + 10V3.

Ans. 5 + 13.

6o. Operations on Imaginary Quantities.

136. An imaginary quantity has been defined to be an indicated even root of a negative quantity.

The rule deduced for multiplying radicals requires some modification, when applied to imaginary quantities. By the rule already deduced, the product of v-4 by ✓ - 3 would be equal to V12; whereas,

the true product is -V12, 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 ✓ – 1; the other factor may be either rational or irrational. Thus,

7–4=2V-1, V-3=V3x V-1, varav -1. The factor, V-1, is called the imaginary factor, and the other one is called its coefficient. Thus, in the expression, V3 V-1, the factor V3 is the coefficient of the imaginary factor ✓– 1.

When several imaginary factors are to be multiplied together, we first reduce them to the form, av – 1. We can then multiply together the coefficients of the imaginary factor by known rules. It remains to deduce a rule for multiplying together the imaginary factors, or what is the same thing, for raising the imaginary factor to a power whose exponent equal to the number of factors.

The first power of V - 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 powers, or -1xV1x V-1

v=1; the fourth power, is the square of the second power, 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.

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To show the use of this table, let it be required to find the continued product of — 4, V – 3, V – 2, V 7, and 8. Reducing these expressions to the proper form, and indicating the multiplication, we have, 2v1 x V3V-1 xV2V-1xViV-1x2v2V 1.

Changing the order of the factors,

(2 X V3 X V2 X V1 x 2V2) (V — 1)5.

Hence, the product is equal to, 8V21 x V-1=8V — 21.

EXAMPLES.

Perform the multiplications indicated below:

1. Vx V = 62. Ans. a x b(V-1)2 = ab. 2. V-a? x V 62 x v.

Ans. abc (V— 1), = - abcv — 1. 3. V = a x V = 72 x v CZ x V = d2.

Ans. abcd (V–1)4

= abcd. 4. (4 + V— 2) (3 – V–2). Ans. 14 - V–2. 5. (2 – V— 2) (2 – V–2). Ans. 2 – 4V — 2. 6. (3 – V — 2) (3 + V – 2).

Ans. 11.

From what precedes, it follows that the only radical parts of any power of an expression of the form, a Łov-1, will be of the form CV - 1.

Properties of Imaginary Quantities.

138. 1o. A quantity of the form, av. – 1, cannot be equal to the sum of a rational quantity and a quantity of the form, bv1

For, if so, let us have the equality,

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squaring both members, we have,

a? = x2 + 26xv=1-62; transposing, and dividing by 26x,

72 - a2 - 22 V-1

26x an equation which is manifestly absurd, for the first member is imaginary, and the second real, and no imaginary quantity can 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 quantity plus a radical of the second degree.

2o. If, a +bV-1 = x + yv — 1, then a = x,

and b = Y

For, by transposition, we have,

bV-1 = (x – a) + yv- 1;

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