Math 114- Intermediate Algebra Integral Exponents & Fractional Exponents (10 )

Math 114- Intermediate Algebra

Integral Exponents & Fractional Exponents (10 )

Exponential Functions

Exponential Functions and Graphs I. Exponential Functions

The function f x( )=a^x, where x is a real number, a > 0, and a ¹ 1, is called the exponential function, base a. Re- quiring the base to be positive would help to avoid the complex numbers that would occur by taking even roots of negative numbers. (Ex. -1, (-1)

1

2, which is not a real number.) The restriction a ¹ 1 is made to exclude the

constant function f x( ) =1^x =1.

Example: f x( )= 5^x, f x( ) ( ) ,= 1 ^x

6 f x( ) ( .= 4 75)^x

*The variable in an exponential function is in the exponent.

II. Graphing an exponential function A. Plotting points

B. Graphing calculator

* Try f x( ) ( )= 1 ^x 2

III. Application

Example: Compound Interest Formula, A P i n

= (1+ )nt

for n compoundings per year: or A P r n

= (1+ )nt

for continuous compounding: A=Pe^rt

1. a total of $12,000 is invested at an annual rate of 9%.

Find the balance after 5 years if it is compounded quarterly:

A P r

n

= ( + )nt = , ( + . =

) ^{( )} , .

1 12 000 1 0 09

4 ^{4 5} 18 726 11 2. a total of $12,000 is invested at an annual rate of 9%.

Find the balance after 5 years if it is compounded continuously:

A=Pe^rt =12 000, e^{0 09 5. ( )} =18 819 75, .

continuous compounding yields interest: 18819.75-18726.11=93.64 IV. The Number e = 2 7182818284. ....

A n n

( ) (= 1+1) ,n

as the n gets larger and larger, the function value gets

closer to e. Its decimal representation does not terminate or repeat; it is irrational. In 1741, Leonard Euler named this number e. You can use the e^x key on a graphing calculator to find values of the exponential

function f x( )=e^x.

Exponents

I. Evaluate exponential expressions.

A. For any positive integer n, aⁿ = × × × × × ×a a a a a n times such that a is the base and n is the exponent

Example: a³ = × ×a a a 3³ = × × = 273 3 3

B. For any nonzero real number a and any integer n, a⁰ =1 And a

a

n n

- = 1

C. In an multiplcation problem, the numbers or expressions that are multiplied are called factors. If a b× = , then a and b are factors of c.c

Examples: a. 6⁰ = b. (-2)⁰ = c. (-2)³ = d. -2³ = e. 4^-2 =

D. Properties of Exponents a^m ×aⁿ =a^{m n+}

a

a a

m n

= ⁽m n^{- )} such that a ¹ 0

(a^m)ⁿ =a^mn (ab)^m =a b^{m m}

( )a b

a b

m m

= m such that b ¹ 0 Examples:

a. (2x^{-2 5}) = b. y^-5 × y² = c. y^-6 ¸ y^-3 = d. (-2x^{3 4}) = e. m^-5 ×m⁵ =

f. (24 )

3

10 8 7

6 3 5

a b c 5

a b c

-

- =

II. Fractional Exponents Definition I.: a^{n n} a

1

= such that n is called the index and a is the radicand.

Definition II: a a a

m

n n m n m

= = ( )

* m is an integer, n is a positive integer, and a is a real number. If n is even, a ³ 0.

Examples: 3 3

1

2 = 3× 3 = 9 4 4

1

3 3

= ³ 4׳ 4׳ 4 =³ 64

12 12

1

2 = ( 12)² = 12× 12 = 144 =12

3 3 3 3 3

1 2

1 2

1 2

1

2 1

× = ⁺ = = 4 4 4 4 4 4

1 3

1 3

1 3

1 3

1 3

1

3 1

× × = ^{+ +} = =

(12 ) 12^{( )( )} 12 12

1 2 2

1

2 2 1

= = = (5 ) 5^{( )( )} 5 5

1 3 3

1

3 3 1

= = =

8 8

8 8 2 2 2 2

2 3 1 3

2 3

1 3

1

3 3

1 3

1

3 3 1

= ^- = =( ) = ^{( )( )} = =

Addition and Subtraction of Polynomials

I. Polynomials in One Variable

a x_n ⁿ +a_n-1x^n-1+....+a x₂ ² +a x₁ +a₀ and that n is a nonnegative integer,

and a_n,...,a₀ are real numbers called coefficients and a_n ¹ 0

A. Definition 1. Terms

2. Degree of the polynomial 3. Leading coefficient

4. Constant term 5. Descending order Examples:

a. 2x⁴ -8x³ + -x 20 b. y² 1 y³ 2 6

- +

6. Monomial

7. Binomial 8. Trinomial

II. Polynomial in Several Variables A. Definition

1. Degree of a term - sum of the exponents of all the variables in that term.

2. Degree of a polynomial - the degree of the term of highest degree.

Examples:

a. 9ab³ -12a b^{2 4} +9

b. 7x y^{4 3} -5x y^{3 2} +3x y² +6

III. Expressions That Are Not Polynomials a. 2 2 5 5

x x

- + x b. 20 - x c. y

y +

+ 1

3 7

IV. Addition and Subtraction of Polynomials

Like terms - terms/expressions that have same variables raised to the same powers.

Combine/collect like terms

Examples:

a. (-5x³ +3x² -x) (+ 12x³ -7x² +3)

b. (8x y^{2 3} -9xy) (- 6x y^{2 3} -3xy)

c. (3x² -2x -x³ +2) (- 5x² -8x -x³ +4)

Multiplication of Polynomials

A. (binomial)(binomial) : binomial multiplied by binomial, use FOIL

(x+4)(x+3)=(x x× ) (+ x×3) (+ 4×x) (+ 4 3× )

F O I L

=x² +3x +4x +12 = x² +7x +12

Examples:

a. (x+5)(x -3)=

b. (2a+3)(a+5)=

c. (x² +3y x)( ² -5y)=

d. (4x +1)² = (4x +1 4)( x+1)=

e. (5x+1)² =(5x +1 5)( x +1)=

f. (3y-2)² =(3y-2 3)( y-2)=

g. (2a-1)² =(2a-1 2)( a-1)=

h. (2a-1 2)( a+1)=

i. (3y-2 3)( y+2)=

j. (5x -1 5)( x +1)=

B. Special Products of Binomials 1. (A +B)² =A² +2AB+B² 2. (A-B)² =A² -2AB+B² 3. (A-B A)( +B)=A² -B²

C. Multiplying Two Polynomials Examples:

a. (a-b)(2a³ -ab+3b²)=

b. (4x y⁴ -7x y² +3y)(2y-3x y² )=

c. (2x +3y+4 2)( x+ y)=

Division of Polynomials and Synthetic Division

I. Divide a Polynomial by a Monomial

Divide each term of the polynomial by the monomial A. Example:

6 8 5

2

6 2

8 2

5

2 3 4 5

2

2 2

x x

x

x x

x

x x x

x

+ -

= + - = + -

B. Try:

1. 4 6 3 5

2

4 3 5 2

2

y x y x y x

xy

- - +

=

2. 6 8 2 x + =

3. 4 2 2 x2 x

x +

4. 5 6 3

3

3 2

y y y

y

+ +

=

5. 4 6 12 8

4

5 4 3 2

2

x x x x

x

- + -

=

6. 6 5 8

3

3 2 3 4 5

2 3

abc a b c ab c ab c

- +

=

7. 3 6 9

6

2 3 5 7

xyz xyz x y z xy

+ -

=

II. Divide a Polynomial by a Binomial

Divide a ppolynomial by a binomial as we perform long division.

A. Examples:

1. x x

x

2 7 10

2

+ +

+ =

2. 6 5 5

2 3

x2 x x

- +

+ =

3. 4 12 3 17

2

2 5

2

x x x

x

- + -

- + =

4. 2 2 19 15

3

3 2

x x x

x

- - +

- =

III. Divide Polynomials Using Synthetic Division

A. When a polynomial is divided by a binomial of the form x- , thea division process can be greatly shortened by synthetic division.

Examples

1. x x x

2 7 10

2

+ +

+ =

2. 6 5 5

2 3

x2 x x

- +

+ =

3. 2 2 19 15

3

3 2

x x x

x

- - +

- =

*Please check against results from 1., 2., and 4. from previous page.

B. Polynomial Division

1. Factor: When we are dividing one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor of the dividend.

P x( )=d x( )×Q x( )+R x( )

P x( ) is the dividend, d x( ) is the divisor, Q x( ) is the quotient, R x( ) is the remainder

Example: Determine

a. Whether x + 1 is a factor of x³ +2x² -5x-6

b. Whether x² +3x - is a factor of x1 ⁴ -81