Bellwork

1. Solve

2. Solve

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Properties of Exponents

Section 6.1 and 6.2

What will we learn?

You Will Learn Use zero and negative exponents.

Use the properties of exponents. Solve real-life problems involving exponents.

Exponential Notation

an = a * a * a * a…* a (where there are n factors)

The number a is the base and n is the exponent.

Zero and Negative Exponents

If a ≠ 0 is any real number and n is a positive integer, thena0 = 1a-n = 1/an

You try

BellworkSimplify the expressions;

Laws of Exponents Product of power property When multiplying two powers of the same base, add

the exponents.aman = am+n

Quotient of power property When dividing two powers of the same base, subtract

the exponents. am/ an = am – n

Power of a power properties When raising a power to a power, multiply the

exponents. (am)n = amn

Example – Product Property

(-5)4 * (-5)5 = (-5)4+5 = (-5)9 = -1953125

Example

x5 * x2 = x5+2 = x7

Example – Neg. Exponent

(-5)-6(-5)4 = (-5)-6+4 = (-5)-2 =

251

251

Example – Quotient of Powers

3

5

xx 35x 2x

Example – Quotient of Powers

10

5

xx 105x 5x 5

1x

Example – Power of a Power

(23)4 = 23*4 = 212 = 4096

Example

(34)2 = 34*2 =38 = 6561

Bellwork

Laws of Exponents (ab)n = anbn

When raising a product to a power, raise each factor to the power.

(a/b)n = an / bn

When raising a quotient to a power, raise both the numerator and denominator to the power.

(a/b)-n = (b/a)n

When raising a quotient to a negative power, take the reciprocal and change the power to a positive.

a-m / b-n = bn / am

To simplify a negative exponent, move it to the opposite position in the fraction. The exponent then becomes positive.

Example – Zero Exponent

(7b-3)2 b5 b = 72 b-3*2 b5 b = 49 b-6+5+1 = 49b0 =49

Example – Power of Quotient

2

5sr

25

2

sr 10

2

sr 102sr

3

yx

3

3

yx

4

7

xx

1

47x 3x

7

5

xx

57

1x 2

1x

Basic Examples

Scientific Notation

Scientific Notation—shorthand way of writing very large or very small numbers.4 x 1013

4 and 13 zero’s1.66 x 10-12

0.00000000000166

Scientific Notation

131,400,000,000= 1.314 x 1011

Move the decimal behind the 1st number

How many places did you have to move the decimal?

Put that number here!

Example – Scientific Notation

131,400,000,000 = 5,284,0001.314 x 1011 =5.284 x 106

61110*284.5314.1 900,2410*249. 5

Bellwork

1. Simplify the following:

2. Simplify the following:

3. Simplify the following:

223 73 xyzzyx

32 238 xyxyxy

3

2

3

35abba

Finding nth RootsYou can extend the concept of a square root to other types of roots.

For example, 2 is a cube root of 8 because = 8, and 3 is a fourth root of 81 because = 81.

In general, for an integer n greater than 1, if = a, then b is an nth root of a. An nth root of a is written as , where the expression n √ a is called a radical and n is the index of the radical. You can also write an nth root of a as a power of a.

nth root

If n is any positive integer, then the principal nth root of a is defined as:

If n is even, a and b must be positive.

means nn a b b a

If you assume the Power of a Power Property applies to rational exponents, then the following is true.

Examples:

36 1. 3612 6

643 2. 6413 4

149

12

3. 4912

149

83

4. 1

8

13 2

Examples:

17

8 13

Rational Exponents

For any rational exponent m/n in lowest terms, where m and n are integers and n>0, we define:

If n is even, then we require that a ≥ 0.

/ nm n ma a

Properties of nth roots

if n is odd

| | if n is even

n n n

nn

n

m n mn

n n

n n

ab a b

a ab b

a a

a a

a a

Rationalizing the Denominator

We don’t like to have radicals in the denominator, so we must rationalize to get rid of it.

Rationalizing the denominator is multiplying the top and bottom of the expression by the radical you are trying to eliminate and then simplifying.

More Examples

43 72 aa 4372 a 714a 232 285 rrr 232285 r 780r

33xy 3333 yx 3327 yx

2

32ba

22

22

32ba

2

2

94ba

3522 nm 3532312 nm 15632 nm 1568 nm

xx

28 4

128 14x 34x

5

3

39zz

35

139z

2

13x 2

3x

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

35abba

323131

313331

35

baba

633

393

35

baba

63

39

27125

baba

36

39

27125ba

3

6

27125ba