Integer Exponents and Scientific Notation

Section 0.2

What’s an exponent?Exponents are shorthand notation for repeated multiplication:

5555 = 54

There are four 5’s being multiplied together.In 54 , the 5 is called the base and the 4 is the power or exponent.

In 5555 , the 5’s are called factors.

Evaluating expressionsEvaluating an expression means to find out what it’s worth (giving it’s value)…just do the math.

(note that the location of the negative sign and the parenthesis make a difference in the answer!)

44

)44(

)4()4(

16

16

16

Evaluate the following:

24

24

24

Evaluating expressions continuedEvaluate the following:

3234

This can become:

333333

or:

36

Which is:

729

This idea is called the Product Property of Exponents. When you are multiplying exponentials with the same base you add the exponents. Just remember the bases MUST be the same.

nmnm aaa

More properties of exponents

4

6

5

5 This can become:5555

555555

Remember that a number divide by itself is 1…

So all that is left is 55 which is 25. This is the Quotient Properties of Exponents. When you divide exponentials with the same base, subtract the exponents.

25

nmn

m

aa

a

More properties of exponents

Power property of exponents:

More properties of exponents

Evaluate numeric expressions

a. (–4 25)2

= 113 = 1331

b. 115

118

–1

= (– 4)2 (25)2

EXAMPLES

10216

16384

8

5

11

11

5

8

11

11

Simplifying Algebraic ExpressionsAlgebraic expressions are simplified when the following things have happened or are “done”:All parenthesis or grouping symbols have been eliminatedA base only appears onceNo powers are raised to other powersAll exponents are positive

EXAMPLES Simplify algebraic expressions

a. b–4b6b7 Product of powers property

b. r–2 –3

s3 ( r –2 )–3

( s3 )–3 = Power of a quotient property

= r 6

s–9Power of a power property

= r6s9 Negative exponent property

c. 16m4n –5

2n–5= 8m4n –5 – (–5) Quotient of powers property

= 8m4n0= 8m4 Zero exponent property

= b–4 + 6 + 7 = b9

EXAMPLE Standardized Test Practice

SOLUTION

(x–3y3)2

x5y6x –6y6

x5y6=

011 yx

11

1

x

The correct answer is B

More Examples 334

223

yx

yx

912

46

yx

yx

4

9126

y

yxx

518 yx

More Examples

4123

322

2

4

zyx

zyx

4462

342

2

4

zyx

zyx

44

4362216

zx

zyyx

1

64 372 zyx

3

7264

z

yx

GUIDED PRACTICESimplify the expression. Tell which properties ofexponents you used.

x–6x5 x3

x2 ; Product of powers property

(7y2z5)(y–4z–1)

7z4

y2

; Product of powers property, Negative exponent property

ANSWER

ANSWER

GUIDED PRACTICE

s 3 2

t–4

s6t8 ; Power of a power property, Negative exponent property

x4y–2 3

x3y6

; Quotient of powers property, Power of a Quotient property, Negative exponent property

x3

y24 ANSWER

ANSWER

Scientific Notation

Scientific Notation was developed in order to easily represent numbers that are either very large or very small. Following are two examples of large and small numbers. They are expressed in decimal form instead of scientific notation to help illustrate the problem

 A very large number:                                             The Andromeda Galaxy (the closest one to our Milky Way galaxy) contains at least 200,000,000,000 stars.                    

A very small number:

On the other hand, the weight of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is0.000,000,000,000,000,000,000,000,006,645 kilograms.As you can see, it could get tedious writing out those numbers repeatedly. So, a system was developed to help represent these numbers in a way that was easy to read and understand: Scientific Notation.

Decimal to Scientific Notation Move the decimal point so the number

shown is between 1 and 10 Count the number of spaces moved and

this is the exponent on the 10 If the original number is bigger than 1, the

exponent is positive If the original number is between 0 and 1,

then the exponent is negative.

What to do for scientific notation

Write in scientific notation: 200,000,000,000

So we write the number in scientific notation as 2.0 x 1011

Write in scientific notation: 0.000,000,000,000,000,000,000,000,006,645

6.645 x 10-27

Scientific Notation to Decimal The number of spaces moved is the

exponent on the 10 Move to the right if the exponent is

positive Move to the left if the exponent is negative

6.45 x 104

2.389 x 10-6

= 64,500

= .000002389