Dealing with Exponents

What do exponents mean

• What does 42 ?

• To multiply 4 by itself 2 times– 4 x 4

• Well what about 4-2? or 45 x 42?

Dealin with inverses Example: 4-2

• A number or letter raised to a negative power means to flip and simplify

• Ex. 3-2= a-3=

• = 53 = y

132

1a3

15-3

1y-1

GUIDED PRACTICE for Examples 3 and 4

2. 7–2 =1

49

3. (–2)–5 =1

32–

When we have the same numbers or letters being multiplied or divided with different exponents we do something different

Multiplication we add the exponentsex. 22 • 24 = 22+4 = 26

EXAMPLE 1 Using the Product of Powers Property

Simplify x4 x7.

= x11

Product of powers property

Add exponents.

x4 x7 = x4 +7

GUIDED PRACTICE for Examples 1

Simplify the expression. Write your answer as a power.

3. a6 a9

= a15

= c16

4. c c12 c3

GUIDED PRACTICE for Examples 1

Simplify the expression. Write your answer as a power.

= 410

= 99

2. 98 9

1. 46 44

EXAMPLE 2 Using the Product of Powers Property

= 33x5

= 27x5

Use properties of multiplication.

Product of powers property

Add exponents.

Evaluate the power.

32x2 3x3

Simplify 32x2 3x3.

= (32 3) (x2 x3)

= 32+1 x2+3

GUIDED PRACTICE for Examples 2

Simplify the expression.

102s4 104s25.

= 1,000,000 s6

GUIDED PRACTICE for Examples 2

Simplify the expression.

63t5 62t86.

= 7776 t13

GUIDED PRACTICE for Examples 2

Simplify the expression.

7x2 7x47.

= 49 x6

GUIDED PRACTICE for Examples 2

Simplify the expression.

= 125 z10

8.

52z 5z7 z2

Simplify

16. (10v5)(-7v6u2)17. (-10e4)(2e5)18. (3i2)(-9i6n2)(-4i6n4)19. (-4m3)(-m2)(7m5v4)20. (9z4)(-8z2)21. (-8a4g6)(4a5)(-9a4)22. (-4t3g6)(-12t2g5)23. (9n5)(-n2g6)24. (-10m6)(-9m2)(12m2r5)25. (11a5)(-12a3e5)26. (10h2)(-5h5)27. (-2t2y3)(9t3y3)(-6t3)28. (-4w6)(-w2s3)29. (8d6)(-6d2b5)30. (10s4e5)(3s2e5)

• 1. (-11m4)(-6m3p2)2. (-2f3)(-3f4s3)3. (10v2)(v2k6)4. (-5l6h4)(10l6h6)5. (-5f2)(f3c3)6. (-12b3z4)(-11b5z5)7. (7w6m2)(9w3m3)8. (-6t5)(-5t6x3)9. (4r5e4)(3r4)10. (2g3)(-3g6u6)11. (-11s4)(-5s3f2)(-2s6)12. (-5o3)(3o6)(-7o2)13. (p6)(8p5n4)14. (-6t4i5)(-10t4)15. (-8b2)(-b6q3)

Dividing Exponents

• When we have division we subtract the two exponents from themselves.

• Subtract from the higher number where its at.

ex. x7

x4 = x7-4

= x3

j2

j5= 1

j5-2

= 1j3

Examples

23

25=

25-3

1

= 122

= 14

3x7

9x4= 3x7-4

9

= x3

3

Simplify

• 1. (7)2 ÷ 74 • 2. 92 ÷ 9-3 • 3. (3)4 ÷ 39 • 4. 58 ÷ 55

• 5. x-3 • x5 • 6. 29 • 2-11

Homework

Scientific Notation

• Is taking a large number and making it to into a more reasonable number.

• Ex. 14500000000000 → 1.45 x 1013

• The Only Rule is the Base number has to be between 1 and 10

EXAMPLE 1 Writing Numbers in Scientific Notation

Stars

There are over 300,000,000,000 stars in the Andromeda Galaxy. Write the number of stars in scientific notation.

SOLUTION

Standard form

300,000,000,000

Scientific notation

3 1011

Product form

3 100,000,000,000

Move decimal point 11 places to the left. Exponent is 11.

ANSWER

The number in scientific notation is 3 1011.

GUIDED PRACTICE for Example 1

Write the number in scientific notation.

SOLUTION

Scientific notation

4 103

Product form

4 1000

1. 4000

Standard form

4000

GUIDED PRACTICE for Example 1

SOLUTION

Standard form

7,300,000

Scientific notation

7.3 106

Product form

7.3 1,000,000

2. 7,300,000

GUIDED PRACTICE for Example 1

SOLUTION

Standard form

63,000,000,000

Scientific notation

6.3 1010

3. 63,000,000,000

Product form

6.3 10,000,000,000

GUIDED PRACTICE for Example 1

SOLUTION

Standard form

230,000

Scientific notation

2.3 105

Product form

2.3 100,000

4. 230,000

GUIDED PRACTICE for Example 1

SOLUTION

Standard form

2,420,000

Scientific notation

2.42 106

Product form

2.42 1,000,000

5. 2,420,000

GUIDED PRACTICE for Example 1

SOLUTION

Standard form

105

6. 105

Product form

1.05 100

Scientific notation

1.05 102

Example 2

• Given: 0.000567• Use: 5.67 (moved 4 places)• Answer: 5.67 x 10-4

• Why did the last slide have the scientific notation to x 10-4

• What definition or rule we make about this

Changing scientific notation to standard form.

To change scientific notation to standard form…

• Simply move the decimal point to the right for positive exponent 10.

• Move the decimal point to the left for negative exponent 10.

(Use zeros to fill in places.)

Example 3

• Given: 5.093 x 106

• Answer: 5,093,000 (moved 6 places to the right)

Example 4

• Given: 1.976 x 10-4

• Answer: 0.0001976 (moved 4 places to the left)

• When multiplying 2 or more scientific numbers we must still follow the rule of Scientific Notation making sure the base is still between 1 and 10

• Do the following