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58 Here 219 16a
is the difference required. 4 28 28 28
2a 6 30 - 46 2. To find the difference of
3a - 46 6ab- 3bb
6ab- 366 – 12ac + 16bc
is the difference required. 12bc
4a 3. Required the difference of and
3a 4. Required the difference of 6e and
5a 20 5. Required the difference of and
3a + c 6. Subtract from
b 2a +6 4a + 8 7. Take from 9
2a 8. Take 2a
from 4a +
To Multiply Fractional Quantities together. MULTIPLY the numerators together for a new numerator, and the denominators for a new denominator *.
1. When the numerator of one fraction, and the denominator of the other, can be divided by some quantity, which is common to both, the quotients may be used instead of them.
2. When a fraction is to be multiplied by an integer, the product is found either by multiplying the numerator, or dividing the denominator by it; and if the integer be the same with the denominator, the numerator may be taken for the product.
a + b
20 1. Required to find the product of g and a X 2a
2a? a? Here
the product required. 8 x 5 40 20
3а 2. Required the product of and
34 7 @ X 3a x 60 18a3
the product required. 3 X 4 X 7 84 14
2a 3. Required the product of
2aa + 2ab
the product required. bx (2a + c) 2ab + bc
4a 4. Required the product of and
462 5. Required the product of and
4 3a 3a
Sac 4ab 6. To multiply
and and together. 62 b
ab 7. Required the product of 2a + and
2a? 262 4a+ 232 8. Required the product of
a + 6
2a 1 9. Required the product of 3a, and 2a + 1
2a +3 22
a? 10. Multiply at
by 2 2a 4a?
Ta Divide one Fractional Quantity by another. Divide the numerátors by each other, and the denominators by each other, if they will exactly divide. But, if not, then invert the terms of the divisor, and multiply by it exactly as in multiplication*.
* 1. If the fractions to be divided bave a common denominator, take the numerator of the dividend for a new numerator, an the numerator of the divisor for the new denominator.
1. Required to divide by
az + ze by
8 8a 2 Here
3a 50 2. Required to divide
2a + b 3a + 2b 3. To divide by
stere, 30-26 4a + 26 2a + b 4a + b 8a + ab + 32 Х
the quotient required. 3a - 26 3a+2b 9a? – 4.62
2a 4. To divide
2a +26 3a?
3ax (a + b) За Here,
Х a + 63
(a3 + b3) xa a2 - abt to is the quotient required.
3.7 11 5. To divide
6x2 6. To divide
3x + 1 4.7 7. To divide by
4.10 8. To divide
4x За 9. To divide
5ac 10. To divide
5a4-564 6a2 + 5ab 11. Divide
4ab + 262 4a 46
2. When a fraction is to be divided by any quantity, it is the same thing whether the numerator be divided by it, or the denominator multiplied by it.
3. When the two numerators, or the two denominators, can be divided by some common quantity, let that be done, and the quotients used instead of the fractions first
INVOLUTION. INVOLUTION is the raising of powers from any proposed root; such as finding the square, cube, biquadrate, &c, of any given quantity. The method is as follows:
MULTIPLY the root or given quantity by itself, as many times as there are units in the index less one, and the last
product will be the power required.-Or, in literals, multiply the index of the root by the index of the power, and the result will be the power, the same as before.
Note. When the sign of the root is +, all the powers of it will be + ; but when the sign is –,
all the even powers will be t, and all the odd powers
- ; as is evident from multiplication.
* Any power of the product of two or more quantities, is equal to the same power of each of the factors, multiplied together.
And any power of a fraction, is equal to the same power of the numerator, divided by the like power of the denominator.
Also, powers or roots of the same quantity, are multiplied by one another, by adding their exponents; or divided, by subtracting their exponents.
Thus, a' x a' @+a'. And a = or
the cubes, or third powers, of x -a and x ta.
EXAMPLES FOR PRACTICE. 1. Required the cube or 3d power of 3a*. 2. Required the 4th power of 2a*b. 3. Required the 3d power of -4a2b3.
a x 4. To find the biquadrate of
262 5. Required the 5th power of a - 28.
6. To find the 6th power of 2a*. Sir Isaac Newton's RULE for raising a Binomial to any
Power whatever * 1. To find the Terms without the Co-efficients. The index of the first, or leading quantity, begins
with the index of the given power, and in the succeeding terms decreases continually by 1, in every term to the last; and in the 2d or following quantity, the indices of the terms are 0, 1, 2, 3, 4, &c, increasing always by 1. That is, the first term will contain only the 1st part of the root with the same index, or of
* This rule, expressed in general terms, is as follows :
In -2 a-x= -12 + n
an222 + n.
n-ln-2 (a-x)"a"-na-), + n. Q9-%%.
333 &c. 2
2 3 Note. The sum of the co-efficients, in every power, is equal to the number 2, when raised to that power. Thus 1+1 = 2 in the first power; 1+2+1=4 = 22 in the square; 1 +3+3 .+1=8 = 23 in the cube, or third power ; and so on.