RATIONAL EXPRESSIONS Chapter 5

5-1 Quotients of Monomials

Multiplication Rule for Fractions Let p, q, r, and s be real numbers with q ≠ 0 and s ≠ 0. Then p r = pr q s qs

Rule for Simplifying Fractions Let p, q, and r be real numbers with q ≠ 0. Then pr = p qr q

EXAMPLES Simplify: Find the GCF of the numerator and denominator.

EXAMPLES GCF = = = 3/4

EXAMPLES Simplify: 9xy 3 15x 2 y 2 Find the GCF of the numerator and denominator.

EXAMPLES GCF = 3xy 2 9x 3 = 3y 3xy 2 15x 2 y 2 5x 3xy 2 = 3y/5x

LAWS of EXPONENTS Let m and n be positive integers and a and b be real numbers, with a ≠ 0 and b≠0 when they are divisors. Then:

Product of Powers a m a n = a m + n x 3 x 5 = x 8 (3n 2 )(4n 4 ) = 12n 6

Power of a Power (a m ) n = a mn (x 3)5 = x 15

Power of a Product (ab) m = a m b m (3n 2 ) 3 = 3 3 n 6

Quotient of Powers If m > n, then a m  a n = a m-n x 5  x 2 = x 3 (22n 6 )/(2n 4 ) = 11n 2

Quotient of Powers If m < n, then a m  a n = 1/a |m-n| x 5  x 7 = 1/x 2 (22n 3 )/(2n 9 ) = 11 n 6

QUOTIENTS of MONOMIALS are SIMPLIFIED When: The integral coefficients are prime (no common factor except 1 and -1); Each base appears only once; and There are no “powers of powers”

EXAMPLES Simplify 24 s 4 t 3 32 s 5

EXAMPLES Answer 3 t 3 4 s

EXAMPLES Simplify -12 p 3 q 4 p 2 q 2

EXAMPLES Answer -3 p q

5-2 Zero and Negative Exponents

Definitions If n is a positive integer and a ≠ 0 a 0 = 1 a -n = 1 a n 0 0 is not defined

Definitions If n is a positive integer and a ≠ 0 a -n = 1/a n

EXAMPLES Simplify: a 0 b -3

EXAMPLES Answer: 2 b 3

5-3 Scientific Notation and Significant Digits

Definitions In scientific notation, a number is expressed in the form m x 10 n where 1 ≤ m < 10 and n is an integer

5-4 Rational Algebraic Expressions

Definitions Rational Expression – is one that can be expressed as a quotient of polynomials, and is in simplified form when its GCF is 1.

EXAMPLES Simplify: x 2 - 2x x 2 – 4 Factor

EXAMPLES Answer: x (x – 2) (x + 2)(x - 2) = x x + 2

Definitions A function that is defined by a simplified rational expression in one variable is called a rational function.

EXAMPLES Find the domain of the function and its zeros. f(t) = t 2 – 9 t 2 – 9 t FACTOR

EXAMPLES Answer: (t + 3) (t – 3) t(t – 9) Domain of t = {Real numbers except 0 and 9} Zeros are 3 and -3

Graphing Rational Functions 1.Plot points for (x,y) for the rational function. 2. Determine the asymptotes for the function. 3.Graph the asymptotes using dashed lines. Connect the points using a smooth curve.

Definitions Asymptotes- lines approached by rational functions without intersecting those lines.

5-5 Products and Quotients of Rational Expressions

Division Rule for Fractions Let p, q, r, and s be real numbers with q ≠ 0, r ≠ 0, and s ≠ 0. Then p  r = p s q s q r

EXAMPLES Simplify. 6xy  3y a 2 a 3 x

EXAMPLES Answer: = 6xy a 3 x a 2 3y = 2ax 2

5-6 Sums and Differences of Rational Expressions

Addition/Subtraction Rules for Fractions 1.Find the LCD of the fractions. 2. Express each fraction as an equivalent fraction with the LCD and denominator 3.Add or subtract the numerators and then simplify the result

EXAMPLES Simplify: = _1 – _1 + _3_ 6a 2 2ab 8b 2 4b2 – 12ab + 9a2 = 24a 2 b 2 = (2b – 3a) 2 24a 2 b 2

5-7 Complex Fractions

Definition Complex fraction – a fraction which has one or more fractions or powers with negative exponents in its numerator or denominator or both

Simplifying Complex Fractions 1.Simplify the numerator and denominator separately; then divide, or

Simplifying Complex Fractions 2.Multiply the numerator and denominator by the LCD of all the fractions appearing in the numerator and denominator.

EXAMPLES Simplify: (z – 1/z)  (1 – 1/z) Use Both Methods

5-8 Fractional Coefficients

EXAMPLES Solve: x 2 = 2x

5-9 Fractional Equations

Definition Fractional equation – an equation in which a variable occurs in a denominator.

Definition Extraneous root – a root of the transformed equation but not a root of the original equation.

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