exponents and exponential functions §5 – 1: quotients of...

Algebra II Name: of 13

CLASS NOTES: §5 – 1, §5 – 2, §10 – 1 and §10 – 2 Exponents and Exponential Functions

§5 – 1: Quotients of Monomials EX

�

x 3 • x 2

EX

�

3x( ) 2

EX

�

x 3⎛ ⎝ ⎜ ⎞

⎠ ⎟

2

EX

�

x

2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3

EX

�

x 8

x 5

EX

�

x 3

x 4

There are MORE x’s on the bottom.

EX 1 Simplify each expression.

(a)

�

9xy 3

15x 2y 2 (b)

�

3x 4

y 2•

y 5

6x 2

Laws of Exponents:

(1)

(2)

(3)

(4)

(5)


EX 2 Simplify each expression.

(a)

�

3x

5y 3

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

(b)

�

5x

4y 2

2y

x 2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3

(c)

�

4x

3y 7

y 4

2 x 2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3

(d)

�

−2 x 3y 2⎛ ⎝ ⎜ ⎞

⎠ ⎟

2

12 x 2y 4

What to do FIRST?????

PEMDAS – Simplify power of a power before multiplying.


§5 – 2: Zero and Negative Exponents

EX

�

2 x 0

EX

�

3x −2

EX

�

a

b

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1

EX 1 Write in simplest form without zero or negative exponents.

(a)

�

1

5x −1 (b)

�

3x( ) −2

(c)

�

3 0 x −3y

2 x −1y −2 (d)

�

2 x −2

5y 3

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1

Laws of Exponents:

(1)

(2)

So….

(3)

To make a negative exponent positive, move the variable to the other side of the fraction bar.


EX 2 Write in simplest form without zero or negative exponents.

(a)

�

x −2y⎛ ⎝ ⎜ ⎞

⎠ ⎟ −1

xy 2⎛ ⎝ ⎜ ⎞

⎠ ⎟ −2

(b)

�

4x −2

y

x 3

3y 2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1

(c)

�

x −1

y −2

x −1

y 2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1

EX 3 Write without using fractions. (Use negative exponents instead!)

(a)

�

1

x 2 (b)

�

2 x

y

(c)

�

5x 2

yz 3 (d)

�

8x 2y

xy 3


§10 – 1: Rational Exponents

This section is about EXPONENTS that are FRACTIONS!

So…

€

x 1 2 = x

x 1 3 = x3

x 1 4 = x4

x 2 3 = x 23

€

x 2 3 = x 23

EX 1 Simplify by rewriting in RADICAL FORM.

(a)

€

163 4 (b)

€

813 2 (c)

€

25 −3 2 (d)

€

92.5

(e) 8−3 5 (f) 91.5

Hint: Rewrite as a fraction.

EXPONENTIAL FORM RADICAL FORM


EXPONENTIAL FORM vs. SIMPLEST RADICAL FORM

SAME ! EX 2 Write the following in EXPONENTIAL FORM.

(a) ab3

(b)

€

a 5b 3

c 23

(c) 1

xy 53

EX 3 Simplify.

(a) 1

43⎛

⎝⎜⎞

⎠⎟

−3 2

(b) 1

813

⎛

⎝⎜⎞

⎠⎟

−3 4

Write using powers…

Write using roots…

Hint: Why might it be helpful to find each of these first??


EX 4 Write

€

8 • 43 …

(a) in EXPONENTIAL form…

(b) in simplest RADICAL form… EX 5 Simplify by going back and forth between exponential and radical forms.

(a) 1

3• 93 (b) 163 • 8

Using EXPONENTS to help SOLVE EQUATIONS.

Put these in exponential form and

use . First you will need the same base.

Rewrite as , and rewrite

as . Then you will have a common base of .

Why? ⇒ If and are reciprocals, then ,

⇒

Strategy: If x is in the base…

• Raise both sides to the reciprocal power.


EX 6 Solve for x by raising both sides to the reciprocal power.

(a)

€

5x −1 3 = 20 ⇐ Isolate

€

x −1 3 ⇐ Raise both sides to the power that is the reciprocal of

€

−1 3 .

(b)

€

x − 1( ) 3 2= 8 ⇐ Raise both sides to the

power that is the reciprocal of

€

3 2 .

(c) 6x 2 3 = 54

(d) t − 4( )2 3 − 3 = 1


§10 – 2: Real Number Exponents Remember the rules of radicals from Chapter 6.

€

xb • yb = x • yb

xb

yb=

x

yb

Also remember that you can only add or subtract similar radicals…

€

xb + xb = 2 xb Real Number Exponents are exponents that can be rational or irrational. Irrational exponents include square roots that cannot be simplified

€

2 , 3 , 5 , ..." # $ %

& ' as well

as the irrational number,

€

π . EX 1 Simplify each expression completely using the laws of exponents.

(a)

€

6 2

6 − 2

(b)

€

4π • 2 3−2 π

⇐ The Product and Quotient Properties apply only if the radicals have the same index “b”.

In order to use the rule:

your terms need to have the same base.

Rewrite as so that both of the

terms have the base of 2.

⇐ Use:


EX 2 Simplify each expression completely using the laws of exponents.

(a)

€

32 2" # $

% & '

2

(b) 32+ 7

32− 7

(c) 6 12 ⋅ 6 27

6 3

(d) 2 + 3( )2+π2 + 3( )π

⇐ Use:


Using EXPONENTS to help SOLVE EQUATIONS.

EX 3 Solve each equation.

(a)

€

8x =1

4

(b)

€

54 −t = 25t −1

(c) 323 = 2x (d) 42x =1

2

⎛

⎝⎜⎞

⎠⎟

1−2x

⇐ Step 1 • In this example, rewrite both

sides as powers of . ⇐ Step 2 • If the bases are equal, then

the exponents are equal.

⇐ Step 1 • In this example, rewrite both

sides as powers of . ⇐ Step 2 • If the bases are equal, then

the exponents are equal.

Strategy: If x is in the exponent…

• Rewrite both sides of the equation so they have the same base.

• Then you can use … (This means if the bases are equal, then the exponents are equal.)


Exponential Functions

The exponential function looks like:

x f x( ) or y = 2x

-1 y = 2−1 =

1

2

0 y = 20 = 1 1 y = 21 = 2 2 y = 22 = 4

Graphs are transformed as follows:

Equation Transformation

y = bx + c Shifts the graph c units up.

y = bx − c Shifts the graph c units down.

y = bx +c Shifts the graph c units left.

y = bx −c Shifts the graph c units right.

y = b −x Reflects the graph across the Reflects the

graph across the y-axis

If and , the function defined by

or

is called the exponential function with base b.


EX 4 Graph the following.

(a) y = 3x and y = 3x +2

(b) y =1

2

⎛

⎝⎜⎞

⎠⎟

x

and y =1

2

⎛

⎝⎜⎞

⎠⎟

x

− 2

exponents and exponential functions §5 – 1: quotients of...

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