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1. It is required to find the product of 38 and 2/6. Here 3√8
1 2 3.5
2. It is required to find the product of and V
18 8 9 8
3. It ́is required to find the product of 2 and 33.
Here *=*= (23)=8&
And›› 3—3*= (32)*—98
4. It is required to find the product of 5/a and 32/a.
Whence 15a=15(a3) †
or 15/a5 Ans.
EXAMPLES FOR PRACTICE.
5. It is required to find the product of 5/8 and 3/5, 6. It is required to find the product of 18 and 5 4.
8. Required the product of 18 and 5/20.
9. Required the product of 2/3 and 135.
10. Required the product of 724a3 and 120*
To divide one surd quantity by another,
When the surds are of the same kind, find the quotient of the rational parts, and the quotient of the surds, and the two joined together, with their common radical sign between them, will give the whole quotient required.
But if the surds are of different kinds, they must be reduced to a common index, and then be divided as before.
It is also to be observed, that the quotient of different powers or roots of the same quantity, is found by subtracting their indices.
1. It is required to divide 8/108 by 2/6.
2/6 4/18-4/9X2=12/2 Ans. 2. It is required to divide 83/512 by 4 3/2. 83/512
3. It is required to divide √5 by √2.
5. It is required to divide 6/54 by 3 √√2. 6. It is required to divide 4/72 by 218.
✔a by 2√ub.
9. It is required to divide 4-✔✔a by 2
10. It is required to divide 32a by 13
11. It is required to divide 9 a by 4- 11
12. It is required to divide
20+√12 by √5√+3. Note. Since the division of surds is performed by subtracting their indices, it is evident that the denominator of any fraction may be taken into the numerator, or the numerator into the denominator, by changing the sign of its index.
= 1, or = am-ma°, it follows, that the
expression ao is a symbol equivalent to unity, and, consequently, that it may always be replaced by 1 whenever it occurs.
(t) To what is above said, we may also farther observe, 1. That added to or subtracted from any quantity, makes it meither greater nor less; that is,
be expressed with a negative index.
6. Let a(a2 —x2) ́1 be expressed with a positive index.
a+0=a, and a—0—ɑ.
2. Also, if nought be multiplied or divided by any quantity; both the product and quotient will be nought; because any number of times 0, or any part of 0, is 0; that is,
Oxa, or a×0, and -=0.
3. From this it likewise follows, that nought divided by nought, is a finite quantity, of some kind or other.
For since 0xa=0, or 0—0Xa, it is evident, that a.
4. Farther, if any finite quantity be divided by 0, the quotient will be infinite.
For let q, then, if b remains the same, it is plain, the less a is, the greater will be the quotient q; whence, if a be indefinitely small, q will be indefinitely great and consequently, when a is 0, the quotient q will be infinite: that is,
Which properties are of frequent occurrence in some of the higher parts of the science, and should be carefully remembered.
To involve, or raise surd quantities to any power.
When the surd is a simple quantity, multiply its index by 2 for the square, by 3 for the cube, &c. and it will give the power of the surd part, which being annexed to the proper power of the rational part, will give the whole power required. And if it be a compound quantity, multiply it by itself the proper number of times, according to the usual rule. (u)
3. It is required to find the square of 33/3.
6. It is required to find the square of 3+2/5.
(u) When any quantity that is affected with the sign of the square root is to be raised to the second power, or squared, it is done by suppressing the sign. Thus,
✔a)2, or ✔a×✔a—a; and(✔a+b)2, or ✅a+bX√a+b=a+b