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Thus a3a3a2
=aa. For a§×a3—a3—a2.
}_a}. Χαι

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287. When radical quantities which are reduced to the same index have RATIONAL CO-EFFICIENTS, THE RATIONAL PARTS MAY BE DIVIDED SEPARATELY, AND THEIR QUOTIENT PREFIXED TO THE QUOTIENT OF THE RADICAL PARTS.

Thus acbdab=cd. For this quotient multiplied into the divisor is equal to the dividend.

I

Divide 24x/ay 18dh√/bx_by(a3x2) 16√32 b√xy

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These reduced to the same index are ab(xb)a and a(x2)a. The quotient then is b(b)*= (b5)3. (Art. 272.)

To save the trouble of reducing to a common index, the division may be expressed in the form of a fraction.

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288. RADICAL QUANTITIES, LIKE POWERS, ARE INVOLVED BY MULTIPLYING THE INDEX OF THE ROOT INTO THE INDEX OF THE REQUIRED POWER.

3 2=a. For a3 Xa+=a3.

1. The square of a3=a3×2_a}.

2. The cube of a=aa×3—aa. For a3×aa×aa—aa.

m

3. And universally, the nth power of aan_a". For the nth power of a*=axa....n times, and the sum of the indices will then be m.

5

4. The 5th power of a y3, is a3y. Or, by reducing the

roots to a common index,

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n

3

(ax)*

And the nth power of a", is aa. That is,

289. A ROOT IS RAISED TO A POWER OF THE SAME NAME, BY REMOVING THE INDEX OR RADICAL SIGN.

Thus the cube of 3√b+x, is b+x.

And the nth power of (a − y)2, is (a—y.)

290. When the radical quantities have rational co-efficients, these must also be involved.

1. The square a"√x, is a2"x2.

For a"√xxa"√x=a3"√x3.

nm

n

2. The nth power of aTM is am.

3. The square of ax-y, is a2 X(x-y.)

4. The cube of 3a3√y, is 27a3y.

×(x—y.)

291. But if the radical quantities are connected with others by the signs and, they must be involved by a multiplication of the several terms, as in Art. 213.

Ex. 1. Required the squares of a+✔y and a-vy.

a+√y

a+ √y

a2+avy

a-vy
a-vy

a2-a√y

avy+y

a2+2a√y+y

2. Required the cube of a-b. 3. Required the cube of 2d+√x.

-a√y+y

a2 −2a√y+y

292. It is unnecessary to give a separate rule for the evolution of radical quantities, that is, for finding the root of a quantity which is already a root. The operation is the same as in other cases of evolution. The fractional index of the radical quantity is to be divided, by the number expressing the root to be found. Or, the radical sign belonging to the required root, may be placed over the given quantity. (Art. 257.) If there are rational co-efficients, the roots of these must also be extracted.

Thus, the square root of a3, is a

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293. It may be proper to observe, that dividing the fractional index of a root is the same in effect, as multiplying the number which is placed over the radical sign. For this number corresponds with the denominator of the fractional index; and a fraction is divided, by multiplying its denomin

ator.

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On the other hand, multiplying the fractional index is equivalent to dividing the number which is placed over the radical sign.

Thus the square of /a or a3, isiya or a

X3

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293.6. In algebraic calculations, we have sometimes occasion to seek for a factor, which multiplied into a given radical quantity, will render the product rational. In the case of a simple radical, such a factor is easily found. For if the nth root of any quantity, be multiplied by the same root raised to a power whose index is n-1, the product will be the given quantity.

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So vax√α=α. And ax/a3 =3/a3=a.

And •√/aו√a3=a, &c. And (a+b)3×(a+b)3⁄4=a+b.

4

And (x+y)1×(x+y)2=

293.c. A factor which will produce a rational product, when multiplied into a binomial surd containing only the squre root, may be found by applying the principle, that the product of the sum and difference of two quantities, is equal to the difference of their squares. (Art. 235.) The binomial itself, after the sign which connects the terms is changed from + to, or from to +, will be the factor required.

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Thus (a+b)×(√a−√/b)=√a2 — √b2=a-b, which is free from radicals.

So (1+2)x(1−√2)=1—2— — 1.

And (3-22)×(3+2√2)=1.

When the compound surd consists of more than two terms, it may be reduced, by successive multiplications, first to a binomial surd, and then to a rational quantity.

Thus (10-2 −√3) ×(√10+√2+√/3)=5−2√6, a binomial surd.

And (5—2/6)×(5+2√6)=1.

Therefore(✓10-2-3)multiplied into(✓10+√2√+3).

X(5+2/6)=1.

293.d. It is sometimes desirable to clear from radical signs the numerator or denominator of a fraction. This may be effected, without altering the value of the fraction, if the

numerator and denominator be both multiplied by a factor which will render either of them rational, as the case may require.

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rational quantity.

Or if both parts of the given fraction be multiplied by ✔x, it will become Vax

2. The fraction

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in which the denominator is rational.

b3 ___ __b3×(a+x)3 _b3× (a+1) 3, (a+x)3 ' (a+x;s+}

3. The fraction Vy+x_(y+x)}+&

4. The fraction

I

α

=

a(y+x) 3}

n- 1

n ах

y+x

a+x

a(y+x) 3

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8×(√3−√2−1)(−√2) √3+√2+1 (√3+√2+1)(√3−√2-1)−√2)

+2√2.

=√5

=4-26

9. Reduce to a fraction having a rational denominator,

10. Reduce

nator,

2
√3
a-√b
a+ √b.

to a fraction having a rational denomi

293 e. The arithmetical operation of finding the proximate value of a fractional surd, may be shortened, by rendering

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