Equations Containing Radicals.

276. An equation containing a single radical may be solved by arranging the terms so as to have the radical alone on one side, and then raising both sides to a power corresponding to the order of the radical.

277. If two radicals are involved, two steps may be

Square,

EXERCISE 106.

Solve:

3

1.2√x + 5 = √28.

8. √3x+7 = 3.

3

3. √x+9=5√x - 3.

4.4=2√x - 3.

2.3√4x-8=√13x-3. 9. 14+ 4x - 40 = 10.

CHAPTER XVIII.

IMAGINARY EXPRESSIONS.

278. An imaginary expression is any expression which involves the indicated even root of a negative number.

It will be shown hereafter that any indicated even root of a negative number may be made to assume a form which involves only an indicated square root of a negative number. In considering imaginary expressions we accordingly need consider only expressions which involve the indicated square roots of negative numbers.

Imaginary expressions are also called imaginary numbers and complex numbers. In distinction from imaginary numbers all other numbers are called real numbers.

279. Imaginary Square Roots. If a and b are both positive, we have

√ab = √a x √δ.

If one of the two numbers a and b is positive and the other negative, it is assumed that the law still holds true; we have, accordingly:

√-4 = √4(-1) = √4 × √– 1 = 2√-1;
√– 5 = √5 (-1) = √5 × √– 1 = 5+ √−1;
a = √a (-1) = √a x √-1=a√-1;

and so on.

It appears, then, that every imaginary square root can be made to assume the form a√-1, where a is a real number.

280. The symbol - 1 is called the imaginary unit, and may be defined as an expression the square of which is - 1.

Hence, √-1 × √- 1 = (√− 1)2 = − 1 ;

√-ax√- b = √a x√- 1 x vô × √– 1

X

281. It will be useful to form the successive powers of

the imaginary unit.

(√-1)

(-1)2

3

= + -1;

= -1;

(-1) = (√-1)2 - 1 = (-1) √– 1 = – √– 1;

4

2

2

(√-1) = (√− 1)2 (√− 1)2 = (-1) (− 1) = + 1;

5

4

(√-1) = (√-1)* √− 1 = (+1) √−1 = + √– 1;

and so on. If, therefore, n is zero or a positive integer,

(√-1)4n+1 = + √−1;

(√-1)4n+2 = -1;
(√-1)4n+3 = – √−1;

(√-1)4n+4= +1.

282. Every imaginary expression may be made to assume the form a + b√-1, where a and b are real numbers, and may be integers, fractions, or surds.

If b = 0, the expression consists of only the real part a, and is therefore real.

If a = 0, the expression consists of only the imaginary part 6-1, and is called a pure imaginary.

283. The form a + b √ - 1 is the typical form of imaginary expressions.

Reduce to the typical form 6+

8.

This may be written 6 + √8 × √-1, or 6+2√2x√-1; here a = 6, and b = 2√2.

284. Two expressions of the form a+b√-1, a-b√-1, are called conjugate imaginaries.

To find the sum and product of two conjugate imagi

From the above it appears that the sum and product of

two conjugate imaginaries are both real.

Since 62 and (c-a)2 are both positive, we have a negative number equal to a positive number, which is impossible.