Section 7.3 Rational Exponents

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Section 7.3

Rational Exponents

Algebra 1

Day 1: Learning Targets

Define a rational exponent.

Define square roots and radical form.

Define and apply the n th

Evaluate n th root.

root expressions.

Rational Exponents

Rational exponents are exponents that contain fractions. We can also rewrite them using radicals ! Try finding the patterns below…use your calculator to evaluate the following!

๏Š

What do you notice???

Square Root Definition

๐‘

1

2

= ๐‘

Practice converting back and forth between exponential form and radical form. Use the formula above to help you!

Square

Root

Definition

What is another way we can think about this?

Recall Exponents:

4

2 =

4

๏‚Ÿ

4

What if we had something that looks like this…

2

1

16 ๐‘œ๐‘Ÿ 16

2

Remember, this is saying that I want to multiply the base number out 2 times or write out the expression using 2 pieces!

Now we know what our answer is from the last slide. However think about what we said to the left. This is asking us to split 16 up into 2 equal pieces…but we only want 1 out of the 2 . Hence the term ROOT when you write it as a radical!

๏Š

How can I break 16 down into 2 equal

4 4

Try some on your own!

#1:

What is

81

1

2

?

# 2:

Find

100

100

1

(or ___)

81 = ___ โˆ™ ___

100 = ___ โˆ™ ___

Thus,

81

1

2

= ___

Thus,

100 = ___

We can also use this same logic for some other problems too!

What if I have something like…

81

1

4

This is now saying that I know 81 can be written as 4 equal pieces and I only want 1 of those 4 pieces.

How can I break 81 down into 4 equal pieces?

___ โ— ___ โ— ___ โ— ___

What would my answer by if I want only

1 out of those 4? ____

Practice with

๐‘› ๐‘กโ„Ž

Roots

#1) What is

27

1

3

?

#2) Find

5

32

27 = __ โˆ™ __ โˆ™ __

32 = __ โˆ™ __ โˆ™ __ โˆ™ __ โˆ™ __

Thus,

27

1

3

= __

Thus, 5

32 = __

What have we been doing every time to rewrite the expression from radical form to exponential form?

๐‘› ๐‘กโ„Ž

Root Definition

1 ๐‘ ๐‘›

= ๐‘› ๐‘

#3) What is

64

1

3

?

#4) Find

3

125

Use the same logic for advanced

๐‘› ๐‘กโ„Ž

roots too!

๏Š

What if I have something like…

16

3

4

This is now saying that I know 16 can be written as 4 equal pieces and I want 3 of those 4 pieces.

How can I break 16 down into 4 equal pieces?

___ โ— ___ โ— ___ โ— ___

What would my answer by if I want 3 out of those 4? ___ โ— ___ โ— ___ = ___

Advanced Root Definition

๐‘ ๐‘š ๐‘›

= ๐‘› ๐‘ ๐‘š

#1) What is

27

2

3

?

27 = __ โˆ™ __ โˆ™ __

Thus,

27

2

3

=

__

โ— __ = __

#2) Find

2

36

3 ๐‘œ๐‘Ÿ ______

36 = __ โˆ™ __

Thus,

36

3

2

=

___ โ— ___ โ— ___ = ___

Try some on your own!

Practice 3:

What is

64

2

3

?

Practice 4:

Find

5

32

2 ๐‘œ๐‘Ÿ ______

How do you feel?

Heads down, thumbs up!

Got it!

Ehh…so, so.

HELP!!!

๏Œ

End of Lesson

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KAHOOT!

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You MUST fill out the exit ticket along with the kahoot to receive participation points for today’s classwork.

Put the exit ticket in the folder that best rates your current understanding of Lesson 7.3!

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HOMEWORK – Day 1 Portion of the Worksheet!

Day 2: Learning Targets

I can solve exponential equations by recognizing how to rewrite expressions in exponential form.

What would x be in the following problems?

We Do: You Do:

Solving Exponential Equations

Power Property of Equality

For any real number ๐‘ > 0 then ๐‘ ๐‘ฅ

= ๐‘ ๐‘ฆ and if and only if ๐‘ ≠ 1 ๐‘ฅ = ๐‘ฆ

.

,

Example 1: If

5 ๐‘ฅ

= 5

Example 2: If

2 ๐‘ฅ+1

3

, then

= 2

7

, ๐‘ฅ = 3 then ๐‘ฅ + 1 = 7

Solving Exponential

Equations

Practice 1:

Solve

6 ๐‘ฅ

= 216

Practice 2:

Solve

25 ๐‘ฅ−1

= 5

6 ๐‘ฅ

= 6

__

Thus, ๐‘ฅ = ___

5 = 5

2 ๐‘ฅ − 1 = 1

2๐‘ฅ − 2 = 1

Thus, ๐‘ฅ =

3

2

Solving Exponential

Equations

Practice #3:

Solve

8 ๐‘ฅ

= 512

Practice #4:

Solve

12

2๐‘ฅ+3

= 144

Homework

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Day 2 Portion of the Worksheet

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Use your time wisely in class because you will also be getting a quiz review worksheet to take home!

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