PROBLEM VIII.

To multiply fractional quantities together,

RULE.*

Multiply the numerators together for a new numerator, and the denominators for a new denominator; and it will give the product required,

* 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 by multiplying the numerator by it; and if the integer be the same with the denominator, the numerator may be taken for the product.

3. When a fraction is to be multiplied by any quantity, it is the same thing, whether the numerator be multiplied by it, or the denominator divided by it.

Multiply the denominator of the divisor by the numera tor of the dividend for a new numerator, and the numerator of the divisor by the denominator of the dividend for a new denominator.

Or, invert the terms of the divisor, and then multiply by it, exactly as in multiplication.

* 1. If the fractions to be divided have a common denominator, take the numerator of the dividend for a new numerator, and the numerator of the divisor for the denominator.

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 der nominator multiplied by it.

3. When the two numerators, or the two denominators, can be divided by some common quantity, that quantity may be thrown out of each, and the quotients used instead of the fractions first proposed.

INVOLUTION.

INVOLUTION is the continual multiplication of a quantity into itself, and the products thence arising are called the powers of that quantity, and the quantity itself is called the root. Or it is the method of finding the square, cube, biquadrate, &c. of any given quantity.

RULE.*

Multiply the quantity into itself, till the quantity be taken for a factor as many times as there are units in the index, and the last product will be the power required.

*Or,

Multiply the index of the quantity by the index of the power, and the result will be the power required.

* Any power of the product of two or more quantities is equal to the same powers of the factors, multiplied together.

And any power of a fraction is equal to the same power of the numerator, divided by the same power of the denominator.

x3 tax

+ax+a2

x*+2ax+a3=square

x + a

x2+2ax2+ a3x

+ ax3+2a3x+a3

x2+3ax2+3u*x+a3 = cube

x + a

**+3ax3+3a2x2 + a3x

+ ax3+3a2x2+3a3x+a*

xa+4ax3 +6a2x2+4a3x+a• = 4th power.

The third power of x is x***, or x.

The fourth power of 2a3b is 24xa13b, or 16a1ś¿☛. The mth power of d" b is a"" bTM.

The second power of ax3 is ax3×2, or axļ3, that is, ax.

And the mth power of a +213 is a3 +x

or

a +x 13, namely, the nth power of the cube root of

NOTE. All the odd powers, raised from a negative root, are negative, and all the even powers are positive.

Thus, the second power of a is -ax-a+a, by the rule for the signs in multiplication.

The third power of a is +a3×—a——a3.

The fourth power is a3x-a=+a*.

The fifth power of a is tax—a——a*, &c.

EXAMPLES FOR PRACTICE.

1. Required the cube of -8x2y3.

Ans-512xy®.