Working With Radicals

24 64

238

4 16

124

Do NowSimplify each of the exponential expressions

2 33 3 233 323

3 2a a 7b b 23a

4

2

3

3

29

3

2

2

34

8

c

c

53 63 63

5a 8b 6a

23 26 122 23

4 12

1 1

c c

You know these rules

2 4 2 4 6a a a a

32 2 2 2 6a a a a a

1 18

62

1 1

a a a a a a a a aa

a aa

Let’s look at how we can work with expressions such as

124

138

1416

324

238

529

4 16 4 29 25 32

3 34 35 32 24 64

Open the TI-NSPIRE File

• Let’s look at some table values for x1/2. • Look for some special values you might

recognize. What would be another way to express x1/2?

• Look at a graph and click on the line to see what graph has been plotted.

Some things your observed

124 4 2

129 9 3

1216 16 4

1236 36 6

How can you rewrite each of the following with a fractional exponent?

49

121

15

c

1249

12121

1215

12c

So we know that

12x x

Open the TI-NSPIRE File

• Let’s look at some table values for x1/3. • Look for some special values you might

recognize. What would be another way to express x1/3?

• Look at a graph and click on the line to see what graph has been plotted.

Some things your observed

1338 8 2

1331 1 1

How can you rewrite each of the following with a fractional exponent?

13125

13216

13513a

13327 27 3

13364 64 4

3 125

3 216

3 5

3 a

So we know that

133x x

Open the TI-NSPIRE File

• Let’s look at some table values for x1/4. • Look for some special values you might

recognize. What would be another way to express x1/3?

• Look at a graph and click on the line to see what graph has been plotted.

Some things your observed

1441 1 1

14416 16 2

How can you rewrite each of the following with a fractional exponent?

14256

1415

14d

14481 81 3

4 256

4 15

4d

So we know that

144x x

1416 2

Which problems can you rewrite?

124

138

1416

324

238

529

4 16 4 29 25 32

3 34 35 32 24 64

4 2 3 8 2 4 16 2

Now let’s find out how we look at

the others!

Open the TI-NSPIRE File

• Let’s look at some table values for 4x, such as 41, 41.5, 42, 42.5, 43, 43.5, 44, etc.

• Look for some special values you might recognize like 41, 42, 43, 44.

• Look at the value for x = 1.5. Remember this is 41.5. What fractional power could replace 1.5?

1 311.5 2 24 4 4

32 8

324To find a value for we need

to do a little math.

13

24

324

13 24

1264 64 8

13

24or

12

3

4

34

3 324 4

Look at x = 2.55

2.5 24 4

52 32

524

15

24

15 24

121024 1024 32

15

24or

5124

54

5 524 4

Try changing7

3.5 24 4

7

16384 128

2 128

724

17

2

17

2

4

4or

1

7 2

712

4

4

12

7

16384

4

7 724 4

Open the TI-NSPIRE File

• Let’s look at some table values for 16x, such as 161, 161.5, 162, 162.5, 163, 163.5, 164, etc.

• Look for some special values you might recognize. First look at 161, 162, 163

• Look at the value for x = 1.25. What fraction could replace 1.25?

51.25 416 16

14

54

1,048,576

16

5416

15

4

15

4

16

16or

14

15 4

5

16

16

4

5

1,048,576 32

2 32

54

54

16

16

5 5

4416 16

Simplify3

1.5 216 16

13 1322 2

13 32

3

1 3

2

3

2

31

16 16 4096 4096 64

16 4

16

16 16 4

16

6or

3 3216 16

So how could we rewrite

74x

71 7

44x x

Simplify each expression:

238

729

1.516

23 8 4

7

9 2187

3 3216 16

Simplify each expression:

23a

72b

1.5c

43d

3 2a 7

b

3c

43d

Simplify each expression:

53a

94b

74c

112d

5 2a 94 b

74 c 11d

Now can you simplify the rest?

124

138

1416

324

238

529

4 164 29 25 32

3 34 35 32 24 64

4 2 3 8 2 4 16 2

34 8 39 27 23 8 4

1416 2

25

2

32

2 4

24

12

9

9 3

33

1

4

4 4

35

3

32

2 8

122464 64

8

Simplify these

12c

13x

14c

32m

23d

52(3 )x

4 z 34 p 25 e

c 3 x 4 c

3m 33x 23d

14z

25e

34p

Using the new ideas to solve equations

3 4x

13 4x

3

13

3 4x

1 64

64

x

x

These statements state that the cube root of some number is 4. What operation will undo the cube root?

We’ll cube both sides.

3 64 4 üOur solution checks

Solve this equation3 4 16c

43 16x

3

4 34 343 416 16x

1 32

8

x

x

These statements are the same. How can we undo raising a number to the 4/3 power?

We’ll raise each side to the ¾ power

üOur solution checks 43 348 8 16

Solve this equation4 3x

14 3x

1 3 3 3 3

81

x

x

These statements are the same. How can we undo raising a number to the 4/3 power?

We’ll raise each side to the power of 4

üOur solution checks4 81 3

4

14

4 3x

Solve this equation3

13

c

13

13

c

127

c

These statements are the same. How can we undo raising a number to the 4/3 power?

We’ll raise each side to the power of 3

üOur solution checks

3

313

13

c

3

1 13

11327