Core Connections Integrated I 8

LAWS OF EXPONENTS AND SCIENTIFIC NOTATION 1.3.1 and 1.3.2 Laws of Exponents

In general, to simplify an expression that contains exponents means to eliminate parentheses and negative exponents if possible. The basic laws of exponents are listed below. (1) xa ! xb = xa+b Examples: x3 ! x4 = x3+4 = x7 27 !24 = 27+4 = 211

(2) xa

xb= xa!b Examples: x10

x4= x10!4 = x6 24

27 = 24!7 = 2!3 or 123

(3) xa( )b = xab Examples: x4( )3 = x4!3 = x12 2x3( )5 = 21!5 ! x3!5 = 25 x15 = 32x15 (4) x0 = 1 Examples: 20 = 1 !3( )0 = 1 1

4( )0 = 1

(5) x!n = 1xn

Examples: x!3 = 1x3

y!4 = 1y4

14!2

= 1142

= 1116

= 16

In all expressions with fractions we assume the denominator does not equal zero. For additional information, see the Math Notes box in Lesson 1.3.2. For additional examples and practice, see the Checkpoint 4 materials.

Example 1 2xy3( ) 5x2y4( ) Reorder: 2 !5 ! x ! x2 ! y3 ! y4 Using law (1): 10x3y7

Example 2

14x2y12

7x5y7

Separate: 147

!"#

$%& '

x2

x5!"#

$%&' y12

y7!"#

$%&

Using laws (2) and (5): 2x!3y5 = 2y5

x3

Chapter 1

Parent Guide with Extra Practice 9

Example 3 3x2y4( )3 Using law (3): 33 ! x2( )3 ! y4( )3 Using law (3) again: 27x6y12

Example 4 2x3( )!2

Using law (5): 1

2x3( )2

Using law (3): 1

22 ! x3( )2

Using law (3) again: 14x6

Example 5

Simplify: 10x7y3

15x!2y3

Separate: 1015

!"#

$%& '

x7

x(2!"#

$%&' y3

y3!"#

$%&

Using law (2): 23x9y0

Using law (4): 23x9 !1= 2

3x9 = 2x

9

3

Problems Simplify each expression. Final answers should contain no parentheses or negative exponents.

1. y5 ! y7 2. b4 !b3 !b2 3. 86 !8"2

4. (y5 )2 5. (3a)4 6. m8m3

7. 12m86m!3 8. (x3y2 )3 9. (y4 )2

(y3)2

10. 15x2y5

3x4y5 11. (4c4 )(ac3)(3a5c) 12. (7x3y5 )2

13. (4xy2 )(2y)3 14. 4x2( )3 15. (2a7 )(3a2 )

6a3

16. 5m3nm5( )3 17. (3a2x3)2(2ax4 )3 18. x3y

y4( )4

19. 6x8y2

12x3y7( )2 20. (2x5y3)3(4xy4 )2

8x7y12 21. x!3

22. 2x!3 23. (2x)!3 24. (2x3)0

25. 5!2 "3 26. 2x3( )!2

Core Connections Integrated I 10

Answers

1. y12 2. b9 3. 84 4. y10 5. 81a4 6. m5 7. 2m11 8. x9y6 9. y2 10. 5

x2 11. 12a6c8 12. 49x6y10

13. 32xy5 14. 64x6

15. a6

16. 125n3m6

17. 72a7x18 18. x12y12

19. x104y10

20. 16x10y5 21. 1x3

22. 2x3

23. 18x3

24. 1

25. 325 26. 9

4x2

Scientific Notation

Scientific notation is a way of writing very large and very small numbers compactly. A number is said to be in scientific notation when it is written as the product of two factors as described below. • The first factor is less than 10 and greater than or equal to 1. • The second factor has a base of 10 and an integer exponent. • The factors are separated by a multiplication sign. • A positive exponent indicates a number whose absolute value is greater than 1. • A negative exponent indicates a number whose absolute value is less than 1. Scientific Notation Standard Form 5.32 × 1011 532,000,000,000 2.61 × 10–15 0.00000000000000261

Chapter 1

Parent Guide with Extra Practice 11

It is important to note that the exponent does not necessarily mean to use that number of zeros. The number 5.32 × 1011 means 5.32 × 100,000,000,000. Thus, two of the eleven decimal places in the standard form of the number are the 3 and the 2 in 5.32. Standard form in this case is 532,000,000,000. In this example you are moving the decimal point eleven places to the right to write the standard form of the number. The number 2.61 × 10–15 means 2.61 × 0.000000000000001. You are moving the decimal point to the left 15 places to write the standard form. Here the standard form is 0.00000000000000261. For additional information, see the Math Notes box in Lesson 1.3.1. Example 1 Write each number in standard form. 7.84 !108 ! 784,000,000 and 3.72 !10"3 ! 0.00372 When taking a number in standard form and writing it in scientific notation, remember there is only one digit to the left of the decimal point allowed. Example 2 Write each number in scientific notation. 52,050,000 ! 5.205 !107 and 0.000372 ! 3.72 !10"4 The exponent denotes the number of places you moved the decimal point in the standard form. In the first example above, the decimal point is at the end of the number and it was moved 7 places. In the second example above, the exponent is negative because the original number is very small, that is, less than 1.

Core Connections Integrated I 12

Problems Write each number in standard form. 1. 7.85 !1011 2. 1.235 !109 3. 1.2305 !103 4. 3.89 !10"7 5. 5.28 !10"4 Write each number in scientific notation. 6. 391,000,000,000 7. 0.0000842 8. 123056.7 9. 0.000000502

10. 25.7 11. 0.035 12. 5,600,000 13. 1346.8

14. 0.000000000006 15. 634,700,000,000,000 Note: On your scientific calculator, displays like 4.35712 (or 4.357E12) and 3.65–3 (or 3.65E–3) are numbers expressed in scientific notation. The first number means 4.357 !1012 and the second means 3.65 !10"3 . The calculator does this because there is not enough room on its display window to show the entire number. Answers 1. 785,000,000,000 2. 1,235,000,000 3. 1230.5 4. 0.000000389 5. 0.000528 6. 3.91!1011

7. 8.42 !10"5 8. 1.230567 !105 9. 5.02 !10"7

10. 2.57 !101 11. 3.5 !10"2 12. 5.6 !106

13. 1.3468 !103 14. 6.0 !10"12 15. 6.347 !1014