RATIONAL EXPRESSIONS

Chapter 5

5-1 Quotients of Monomials

Multiplication Rule for FractionsLet p, q, r, and s be real numbers with q ≠ 0 and s ≠ 0. Then

p •r = pr q s qs

Rule for Simplifying FractionsLet p, q,and r be real numbers with q ≠ 0. Then

pr = p qr q

EXAMPLES

Simplify:30

40Find the GCF of the numerator and denominator.

EXAMPLES

GCF = 1030 = 10 • 3

40 10 • 4

= 3/4

EXAMPLES

Simplify:9xy3

15x2y2

Find the GCF of the numerator and denominator.

EXAMPLES

GCF = 3xy2

9x3 = 3y • 3xy2

15x2y2 5x • 3xy2

= 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

•am • an = am + n

•x3 • x5 = x8

•(3n2)(4n4) = 12n6

Power of a Power

•(am)n = amn

•(x3)5 = x15

Power of a Product

•(ab)m = ambm

•(3n2)3 = 33n6

Quotient of Powers

If m > n, thenam an = am-n

x5 x2 = x3

(22n6)/(2n4) = 11n2

Quotient of PowersIf m < n, thenam an = 1/a|m-n|

x5 x7 = 1/x2

(22n3)/(2n9) = 11 n6

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

Simplify24s4t3

32s5

EXAMPLES

Answer3t3

4s

EXAMPLES

Simplify-12p3q

4p2q2

EXAMPLES

Answer-3p

q

5-2 Zero and Negative Exponents

Definitions

If n is a positive integer and a ≠ 0

a0= 1 a-n = 1

an

00 is not defined

Definitions

If n is a positive integer and a ≠ 0

a-n = 1/an

EXAMPLES

Simplify:-2-1a0b-3

EXAMPLES

Answer: -1

2b3

5-3 Scientific Notation and

Significant Digits

DefinitionsIn scientific notation, a number is expressed in the form m x 10n where

1 ≤ m < 10 and n is an integer

5-4 Rational Algebraic

Expressions

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

EXAMPLES

Simplify:x2 - 2x

x2 – 4

Factor

EXAMPLESAnswer:

x (x – 2) (x + 2)(x - 2)

= xx + 2

DefinitionsA 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) = t2 – 9 t2 – 9t FACTOR

EXAMPLESAnswer:

(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 a2 a3x

EXAMPLESAnswer:

= 6xy •a3x a2 3y

= 2ax2

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

EXAMPLESSimplify:

= _1 – _1 + _3_ 6a2 2ab 8b2

4b2 – 12ab + 9a2

= 24a2b2

= (2b – 3a)2

24a2b2

5-7 Complex Fractions

DefinitionComplex 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:

x2 = 2x + 1 2 15 10

5-9 Fractional Equations

DefinitionFractional equation

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

DefinitionExtraneous root – a

root of the transformed equation but not a root of the original equation.

END