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10. Multiply V + 2Vx + 4, by Vx + 2√x.

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13.

2+ + 2x + +

x

+ 2x1

+

+4

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+

11. (a2 + a*x*+ a‡x + x3) × (a‡ — x*).

4x

4x + 8x+

+ 8 x 3

These results are identical, but the second has been obtained by following the ordinary rule for exponents; hence, we conclude that the rule for multiplication is the same whether the exponents are entire or fractional.

8x3 + 8x+

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ax2

b8

3

6

3

6

14. (Va++ Vato) × (Va+_ Va+i).

X

d'

3

Ans. Var - Vato.

1

1

15. ( + {} √ @2 = ~) × ( + {} √ @ — @).

g

X

2

a2

Ans. + 01

4

16. (

(8xa + 2x‡y + ‡xły2+ ży3) × (2x+ − 1).

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4°. Division of Radicals.

133. Let a and c represent any two radicals, after having been reduced to a common index. The quotient of the first by the second may be represented as follows:

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RULE.

b

a2 — c2.

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Ans. 16x

; hence,

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Reduce the radicals to a common index; then divide the coefficient of the dividend by that of the divisor for a new coefficient, and the quantity under the radical sign in the dividend by that in the divisor for a new quantity under the radical sign, leaving the index unchanged.

1.

Perform the following indicated divisions:

2

13 1

3

3

W/} + H{V}

3

3

4

5'

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EXAMPLES.

1 1

3

2

√2bx.

5. 2√2ax V√4bx2.

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27

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Ans.

8 1 1

6

FIRST OPERATION.

Ans. 28a3x3 ÷ √16b2x2 = 2.

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28 3 10

39

Ans.

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Vxys

√xy + √xy3 + y

√xy + √xy3 + Y

0

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By combining the above rule with that for the division of polynomials, any complicated radical expression may be divided by another.

9. Divide x + √xy + y, by √x + Vxy + √ỹ.

3

a

6 a3
262x

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x + x3y3 + y

x + xÏy‡ + x3y3 | x

SECOND OPERATION.

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These results are identical; but the second one has been obtained by following the ordinary rule for exponents. Hence, we conclude that the operation for division is the same, whether the exponents are entire or fractional.

y

16

− xfył + y

– xłył – xłył – xłył

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xžy‡ + x1y + y
x*y* + xły2 + y

0

(2x+ — 2).

Ans. 8x + 2xy + ‡x‡y2 + ‡y3.

5o. Reduction of Radicals.

α

a

134. It is often desirable to transform radical expressions of the form and into √b + √c No-Ne equivalent expressions, in which the denominator is rational, that is, which does not contain any radical.

The first form may be thus transformed, by multiplying both terms by b-ve; and the second, by multiplying both terms by √ō + √ē, giving

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a√ī — a√c b с If only one term of the denominator contains a radical, the same rule will hold good.

and

a√b + a√c

b

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с

EXAMPLES.

Render the denominators of the following fractions

rational:

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Va

α

Ans.

a + √o

44 +81/3

109

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= x + y

Ans.

Ans.

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=

y

24-31/2 12 3
62

31

62

44

109

2√3-4 3-4

3√/2+3√3

2

3

=

Ans. √2 - √3 + √/6 − 2.

6. 4 ÷ (√5 + 1).

Ans. 51.

7. (√a + x + √ a − x) ÷ (√ a + x − √ a − x).

a

a2

Ans. +

+

135. In the solution of certain equations, it often becomes necessary to extract the square root of expressions of the form, a + √b, and a√b. In some cases, this operation may be performed, in other cases it cannot be performed. To investigate a rule for determining when the operation can be performed, and the manner of performing it, assume

8

109

√3.

√2.

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