Bell Ringer

2. 7(x + 3)= 105 7x + 21 = 105 -21 -21 7x = 84 7 7 x = 12

Solve.1. 5x + 18 = -3x – 14 +3x +3x 8x + 18 = -14

- 18 -18 8x = -32

8 8 x = -4

Quiz Results

Since there are still a few who haven’t taken the quiz, I’ll give out the results as soon as they do.

If you want to know your grade, log onto your Gradebook and check it yourself.

Otherwise, you’ll have to wait… NO, I’m not digging through the papers

to tell you your grade.

Exponents and Radicals

NCP 503: Work with numerical factorsNCP 505: Work with squares and square roots of numbersNCP 506: Work problems involving positive integer exponents*NCP 504: Work with scientific notationNCP 507: Work with cubes and cube roots of numbersNCP 604: Apply rules of exponents

Basic Terminology

34

Exponent

Base

= 3•3•3•3 = 81

The base is multiplied by itself the same number of times as the exponent calls for.

Its read, “Three to the fourth power.”

Important Examples

-34 = –(3•3•3•3) = -81

(-3)4 = (-3)•(-3)•(-3)•(-3) = 81

(-3)3 = (-3)•(-3)•(-3) = -27

-33 = –(3•3•3) = -27

Variable Expressions

x4 = x • x • x • x y3 = y • y • y

Evaluate each expression if x = 2 and y = 5

x4 y2 = (2•2•2•2)•(5•5) = 4003xy3 = 3•2•(5•5•5)

= 750

Zero Exponent PropertyNegative Exponent Property

Product of PowersQuotient of Powers

Laws of Exponents, Pt. I

Zero Exponent Property

Any number or variable raised to the zero power is 1.

x0 = 1 y0 = 1 z0 = 1

70 = 1 -540 = 1 1230 = 1

Negative Exponent

Any number raised to a negative exponent is the reciprocal of the number.

x-1 = y-1 = 5-1 =

x-2 = 3-2 = = 5-3 = =

1X

1y

15

1X2

132

19

153

1 .125

Negative Exponent

3x-3 = 5y-2 =

2x-2 y2= 3-2 x4=

3x3

5y2

2y2

x2 x4

32

Only x is raised to the -3 power!

Only x is on the bottom.

x4

9 =

Product of Powers

This property is used to combine 2 or more exponential expressions with the SAME base.

53•52 = (5•5•5)•(5•5)

= 55

If the bases are the same, add the exponent!

x4•x3 = (x•x•x•x)•(x•x•x) = x7

Multiplication NOT Addition!

Product of Powers

6-2•6-3

=

165

162•63

=

=

17776

x-5•x-7

=

1x12

1x5•x7

=

n-3•n5 =

n2 n-3+5 =

Product of powers also work with negative exponents!

Quotient of Powers

This property is used when dividing two or more exponential expressions with the

same base.

x6

x3= x6-

3

= x3

Subtract the exponents! (Top minus the bottom!)

Quotient of Powers

67

65= 67-

5

= 62

x3

x5= x3-

5

= x-2 =

1x2

x3

x5= x ∙ x ∙

xx∙x∙x∙x∙x

=OR

1x2

= 36

Laws of Exponents, Pt. II

Power of a PowerPower of a ProductPower of a Quotient

Power of a Power

This property is used to write an exponential expression as a single power of the base.

(63)4 = 63•63•63•63

= 612

When you have an exponent raised to an exponent, multiply the

exponents!

(x5)3 = x5•x5•x5 = x15

Power of a Power

(54)8 = 532

(n3)4 = n12

(3-2)-3 = 36

(x5)-3 = x-15

Multiply the exponents!

= 1x15

Power of a Product

(xy)3

(2x)5

(xyz)4

Power of a Product – Distribute the exponent on the outside of the parentheses to all of the terms inside of the parentheses.

= x3y3

= 25 ∙ x5

= 32x5

= x4 y4 z4

Power of a Product

(x3y2)3

(3x2)4

(3xy)2

More examples…

= x9y6

= 34 ∙ x8

= 81x8

= 32 ∙ x2 ∙ y2

= 9x2y2

Power of a Quotient

Power of a Quotient – Distribute the exponent on the outside of the parentheses to the numerator and the denominator of the fraction.

xy

=( )5 x5

y5( )

Power of a Quotient

More examples…

2x

=( )3 23

x3( )=8x3

3x2y

=( )4 34

x8y4( )= 81x8y4

Basic Examples

Basic Examples 32 xx 32x 5x

34x 34x 12x

3xy 33yx

3

y

x3

3

y

x

4

7

x

x 3x

7

5

x

x2

1

x

Basic Examples

More Difficult Examples

More Examples

43 72 aa 4372 a 714a

232 285 rrr 232285 r 780r

33xy 3333 yx 3327 yx

2

3

2

b

a

22

22

3

2

b

a2

2

9

4

b

a

3522 nm 3532312 nm 15632 nm 1568 nm

x

x

2

8 4

12

8 14x 34x

5

3

3

9

z

z 35

1

3

9

z

2

13x 2

3

x

More Examples

223 73 xyzzyx 21121373 zyx 33421 zyx

32 238 xyxyxy 312111238 yx 6348 yx

22232 23 xyyx 222121232221 23 yxyx

4264 49 yxyx 462449 yx 10636 yx

3

2

3

3

5

ab

ba

323131

313331

3

5

ba

ba

633

393

3

5

ba

ba

63

39

27

125

ba

ba

36

39

27

125

b

a3

6

27

125

b

a