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230. To extract the Square Root of a Binomial Surd.

Extract the square root of a + √б.

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From these two values of x and y, it is evident that this method is practicable only when ab is a perfect square.

(1) Extract the square root of 7+ 4√3.

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227. The sum or difference of two dissimilar quadratic surds cannot be a rational number, nor can it be expressed as a single surd.

For if Va√b could equal a rational number c, we should have, by squaring,

that is,

a± 2√ab+b= c2;

+

2√ab = c2 - a - b.

Now, as the right side of this equation is rational, the left side would be rational; but, by § 226, Vab cannot be rational. Therefore, Va±√b cannot be rational.

In like manner, it may be shown that √a±√b cannot be expressed as a single surd Vc.

228. A quadratic surd cannot equal the sum of a rational number and a surd.

For if a could equal c+√b, we should have, by squaring,

a=c2+2c√b + b,

and, by transposing,

2c√ba-b-c2.

That is, a surd equal to a rational number, which is impossible.

229. If a +√b=x+√ÿ, then a will equal x, and b will equal y.

For, by transposing, √b-√y=x-a; and if b were not equal to y, the difference of two unequal surds would be rational, which by § 227 is impossible.

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230. To extract the Square Root of a Binomial Surd.

Extract the square root of a + √ō.

√a + √b = √x + √y.

Suppose

(1)

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From these two values of x and y, it is evident that this method is practicable only when ab is a perfect square.

(1) Extract the square root of 7+ 4√3.

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A root may often be obtained by inspection. For this purpose, write the given expression in the form a + 2√b, and determine what two numbers have their sum equal a, and their product equal b.

(2) Find by inspection the square root of 18+ 2√77.

It is required to find two numbers whose sum is 18 and whose product is 77; and these are evidently 11 and 7.

Then 18 + 2√77 = 11 + 7 + 2√11 × 7, = (VII+ √7)2.

That is, √11+ √7 = square root of 18 + 2√77.

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(3) Find by inspection the square root of 75 12√21. It is necessary that the coefficient of the surd be 2; therefore, 75-12√21 must be put in the form

75-2√756.

The two numbers whose sum is 75 and whose product is 756 are 63 and 12.

Then

That is,

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= (√63 — √12)2 = (3√7 – 2√3)2. 3√7-2√3 = square root of 75 - 12√21.

Exercise 92.

Extract the square roots of:

7. 9-6√2.

1. 14+6√5. 6. 20-8√6.
2. 17+4√15.
3. 10+2√21. 8. 94-42√5.
4. 16+2√55. 9. 13-2√30.

5. 9-2√14. 10. 11-6√2.

16. 2a+2√a2 — b2.
17. a2-2b√a2 -- b2.

11. 14-4√6.

12. 38 1210.
13. 103-12√11.
14. 5712√15.
15. 31-√10.

18. 8712√42.
19. (a+b)2-4(a—b)√ab.

CHAPTER XVIII.

IMAGINARY EXPRESSIONS.

231. 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.

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

I. √ab = √ax √ō.

II. (√a)2 = a.°

If one of the two numbers a and b is positive and the other negative, law I. is assumed still to apply; we have, accordingly:

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

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