Dealing with Exponents
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