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10. Multiply Px + 2x + 4, by Yu + 2Vc.

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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.

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12.

(= +*+Va+)* (*+- Va+?)

+

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133. Let añī and cķā 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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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.

EXAMPLES.

Perform the following indicated divisions :

1

3

28

1.

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13
4

Ans.

3 10 3

3

39

2.

Ans.

2

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4 25

7 y 30

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VI+
V +/
V20x =

VA
Vi+iv.am VW+V=V

2V

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4.

1 2

8 3:

5. 2V 2ax : 4622.

Ans. 2V8a323 166224 = 2

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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 + Väy + y, by Vã + xy + Vy.

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These results are identical; but the second one has been ob. tained 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.

7.

(162 – 1 )

(2.3– %).
Ans. 8* + 2xły + 3x+ya + {ys.

5o. Reduction of Radicals.

a

a

134. It is often desirable to transform radical expressions of the form

and

into Vīt '

6 c” 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 Vī – Vc; and the second, by multiplying both terms by + Vē, giving avā - arc avá +ave.

and

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If only one term of the denominator contains a radical, the same rule will hold good.

EXAMPLES. Render the denominators of the following fractiong rational:

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5. (V3 – V2) - (V2 + 1).

Ans. V2 – 13 + V6 — 2. 6. 4: (V5 + 1).

Ans. V5 – 1. 17. (Va + x + va — x) = (Va + 2 – Va

Na — «).

a? Ans.

1. 22

a

+

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135. In the solution of certain equations, it often becomes necessary to extract the square root of expressions of the form, a + V6, and a Vī.

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 Vat Vo = x + y

(1)

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