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PROBLEM VI. To find the Difference of Radical Quantities. (152.) It is evident that the subtraction of radical quantities may be performed in the same manner as addition, except that the signs in the subtrahend are tɔ be changed according to Art. 48. Hence we have the following

RULE. When the radicals are similar, subtract their coefficients, and to the difference annex the common radical.

But if the radicals are not similar, and can not be made similar, their difference can only be indicated by the sign of subtraction.

Quest.–Give the rule for the subtraction of radical quantities.

Ex. 1. Find the difference between 5m va and 2m va.

Ans. 3m va. Ex. 2. Find the difference between 7ab v2 and 3ab v2.

. Ans. 4ab v2. Ex. 3. Find the difference between V75 and 127

Ans. 273. Ex. 4. Find the difference between V 150 and v24.

Ans. 376. Ex. 5. Find the difference between 448 and v112.

Ans. 477. Ex. 6. Find the difference between 5720 and 3 / 45.

Ans. V5. Ex. 7. Find the difference between 250 and 18.

Ans. 7V2. Ex. 8 Find the difference between 180a*x and V 20a-x.

Ans. 2av5%. Ex. 9. Find the difference between 2 V72a2 and ✓162a.

Ans. 3a v2. Ex. 10. Find the difference between V490amo and ✓ 40am'.

Ans. 5m v 10a..

PROBLEM VII. (153.) To multiply Radical Quantities together. Let it be required to multiply va by vb. The product, vax vb, will be Vab. .

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For if we raise each of these quantities to the second power we obtain the same result, ab; hence these two expressions are equal. We therefore have the following

RULE. Multiply the quantities under the radical signs together, and place the radical sign' over the product.

If the radicals have coeficients, these must be multiplied together, and the product placed before the radical sign. Ex. 1. What is the product of 3 V8 and 2 V6?

Ans. 6 V 48 which equals 6 v16x3, or 24 v3. Ex. 2. Whát is the product of 578 and 35?

Ans. 15 v40 or 30 v10. Ex. 3. What is the product of 2v3 and 375?

Ans. 6 v 15. Ex. 4. What is the product of 2 v 18 and 3 V20 ?

Ans. 6 v360 or 36 v10. Ex. 5. What is the product of 572 and 3v8?

Ans. 15/16 or 6C Ex. 6. What is the product of 2 v3ab and 3 v2abi

Ans. 6 v6ab or 6ab v6. Ex. 7. What is the product of 7 V5 and 5 V15 ?

Ans. 35 V75 or 175 v3. Ex. 8. What is the product of 2b Vay and 56 vxy?

. Ans. 106ʻxy. Ex. 9. What is the product of 2 V ab and 5 V bc ?

Ans. 10 vabc or 10abc.

QUEST.–Give the rule for multiplying radical quantities

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Ex. 10. What is the product of 2a v 15a and 3a V3a?

Ans. 6a V 45aor 18ao V5.

Vai-va.

PROBLEM VIII. (154.) To divide one Radical Quantity by another. Let it be required to divide va' by va'. '

The quotient must be a quantity which, multiplied by the divisor, shall produce the dividend. We thus obtain va; for, according to Art. 153,

VaRx varva'; that is, Hence we have the following

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

If the radicals have coefficients, divide the coeffi. cient of the dividend by the coefficient of the divisor, and place the quotient before the radical sign. Ex. 1. Divide 8 v108 by 276.

Ans. 4 V18 or 479x2, or 12 V2. Ex. 2. Divide 4 v6a'y by 2 v3y.

Ans. 2 V2a or 2a V2. Ex. 3. Divide v 10aby v5.

Ans. V2a or a v2. Ex. 4. Divide 4aby21 by v7.

Ans. 4ab v3.

Quest.-Give the rule for the division of radical quantities

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PROBLEM IX. To Extract the Square Root of a Polynomial. (155.). In order to discover a method for extracting the square root of a polynomial, let us consider the square of a+b, which is a'+2ab+bo. If we write the terms of the square in such a manner that the powers of one of the letters, as a, may go on continually decreasing, the first term will be the square of the first term of the root; and since, in the present case, the first term of the square is a', the first term of the root. must be a.

QUEST.- Explain the method of extracting the square root of a poly. nomial.

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