18. From the principle for multiplying fractions we have: A power of a fraction is a fraction whose numerator is the like power of the numerator of the given fraction, and whose denominator is the like power of the denominator; or, stated symbolically,

The converse of the principle evidently holds; that is,

Express each of the following fractions as the square of a fraction:

19. The Reciprocal of a fraction is a fraction whose numerator is the denominator, and whose denominator is the numerator of the given fraction.

20. The product of a fraction and its reciprocal is 1.

21. The quotient of one fraction divided by another is equal to the product of the dividend and the reciprocal of the divisor; or, stated symbolically,

+ 2 = · (a ÷ b) ÷ (c ÷ d), by definition of a fraction,

= a ÷ b÷c xd, since ÷ (c÷ d) = ÷ c × d,
= a ÷ b× d÷c, since cx d = × d÷c,
= (a ÷ b) × (d÷c), since x d ÷ c = × (d ÷ c),
by definition of a fraction.

= X

22. If the numerator and denominator of the dividend be multiples of the numerator and denominator of the divisor, respectively, the following principle should invariably be used:

The quotient of one fraction divided by another is a fraction whose numerator is the quotient of the numerator of the first fraction divided by the numerator of the second, and whose denominator is the quotient of the denominator of the first fraction divided by the denominator of the second; or, stated symbolically,

810

+ =
2 = (a + b) ÷ (c ÷ d), by definition of a fraction,

= a ÷ b÷c × d, since ÷ (c÷d) = ÷ c × d,
= a ÷ c÷b × d, since÷b ÷ c = ÷ c ÷ b,
= (a ÷ c) ÷ (b ÷ d), since ÷ b × d = ÷ (b ÷ d),

23. Observe that a fraction is divided by an integral expression, which is a factor of its numerator, by dividing its numerator by the expression.

Also that a fraction is divided by an integral expression, which is not a factor of its numerator, by multiplying its denominator by the expression.

Complex Fractions.

24. A Complex Fraction is a fraction whose numerator and denominator, either or both, are fractional expressions.

Observe that the line which separates the terms of the complex fraction is drawn heavier than the lines which separate the terms of the fractions in its numerator and denominator.

If no distinction be made between the lines of division, the indicated divisions are to be performed successively from above downward.

E.g., 1=2÷3÷4÷5=2÷(3×4×5), by Ch. ́II., § 4, Art. 8, =2÷60 = 3;

25. Complex fractions are simplified by applying successively the principles already established for simple fractions.

We might have simplified the complex fraction in Ex. 2 by first multiplying both its terms by mn, the L. C. D. of the fractions in them, instead of uniting these fractions before