Laws of Exponents

Objective:TSW simplify powers.TSW simplify radicals.

TSW develop a vocabulary associated with exponents.

TSW use the laws of exponents to simplify.

Exponents The lower number is called the base and the

upper number is called the exponent.

The exponent tells how many times to multiply the base.

Exponents

73

exponent

base

power

1. Evaluate the following exponential expressions:

A. 42 = 4 x 4 = 16 B. 34 = 3 x 3 x 3 x 3 = 81 C. 23 = D. (-1) =7

Squares To square a number, just multiply it by itself.

                  

   

=

= 3 x 3 = 93 squared =

Perfect Squares 1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49

8² = 64 9² = 81 10² = 100 11² = 121 12² = 144 13² = 169

Square Roots A square root goes the other direction.

3 squared is 9, so the square root of 9 is 3

3 9

Square Roots11

4 2

9 3

16 4

25 5

36 6

749

864

981

10100

11121

12144

13691

Radicals- The inverse operation of raising a number to a power.

For Example, if we use 2 as a factor with a power of 4, then we get 16. We can reverse this by finding the fourth root of 16 which is 2.

= 216

4

Radicals In this problem, the 16 is called the radicand,

the 4 is the index, and the 2 is the root. The symbol is known as the radical sign. If

the index is not written, then it is understood to be 2.

The entire expression is known as a radical expression or just a radical.

Example Simplify:

a) c)

b) d)

8127

16 8

34

3

Laws of Exponents Whenever we have variables which contain

exponents and have equal bases, we can do certain mathematical operations to them.

Those operations are called the “Laws of Exponents.”

Laws of Exponents

m

mmmnnm

mmmnmnm

yx

yxxx

yxxyxxx

.4.3

.2.1

Laws of Exponents

mnn

m

nmn

m

xxxthenmnifb

xxxthennmifa

1,.5

,.5

Zero Exponents

A nonzero based raise to a zero exponent is equal to one

a0 = 1

Negative Exponents

a-n= ( 1

______

an )

A nonzero base raised to a negative exponent is the reciprocal of the base raised to the positive exponent.

Basic Examples

32 xx 32x 5x

34x 34x 12x

Basic Examples

3xy 33yx

3

yx

3

3

yx

4

7

xx

1

47x 3x

7

5

xx

57

1x 2

1x

Basic Examples

Examples

5

pu

74 xx

57y

3

9

xx

1.

2.

3.

4.

Scientific NotationObjective:

TSW rewrite numbers in scientific notationTSW perform operations with numbers in

scientific notation.TSW solve real-world problems using

numbers in scientific notation.

When we multiply a number by a positive power of 10, we move the decimal point to the right the number of places indicated by the exponent.

This method of numbers is known as scientific notation.

When we write a number greater than or equal to ten in scientific notation, we use three steps: 1. place the decimal point just right of the

first nonzero digit 2. count the number of places the decimal

point moved to the left 3. multiply the number in step one by 10ª (a

is the number of places the decimal point moved) to indicate where the decimal point should be.

Example

Write 7,024,000 in scientific notation

Example

Write 476.23 in scientific notation.

We can also write very small numbers in scientific notation.

For these, we use negative exponents.

We use 10 with a negative exponent to show that the decimal point should be moved to the left.

When we write a number between zero and one in scientific notation, we use three steps:

1. place the decimal point just to the right of the first nonzero digit

2. count the number of places the decimal point moved to the right

3. multiply the number in step one by 10ˉª (a is the number of places the decimal point moved) to indicate where the decimal point should be

Example

Write 0.0652 in scientific notation.

Example

1. Write these numbers in standard notation:a.) 4.6 x 10ˉ³

b.) 4.6 x 10

2. Saturn is about 875,000,000 miles from the sun. What is this distance in scientific notation?

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Answers

1. a.) 0.0046

b.) 4600000

2. 8.75 x 108

Computing with Scientific Notation You can multiply and divide numbers written

in scientific notation. (Use the Laws of Exponents!) To multiply powers with the same bases, add the

exponents To divide powers with the same base, subtract the

exponents

(3.2 x 10²) x (2 x 10³) Step 1: Multiply the first pair of factors from

each (3.2 x 2) = 6.4

Step 2: Multiply the second pair of factors ( the ones written in exponential form)

10² x 10³ = 10 = 10 Step 3: Combine the products

6.4 x 10

2 + 3 5

5

Examples 1. (5.4 x 10 ) x (4.6 x 10³)

2. (8.4 x 10³) x (2.1 x 10 )

3. (1.2 x 10 ) x (9.6 x 10²)

5

-4

-4

Answers 1. 2.484 x 10

2. 4 x 10

3. 1.25 x 10

9

7

-7

Adding and Subtracting

You can also add and subtract with numbers written in scientific notation as long as the second factors are the same.

Example About 8.73 x 10 people in the world speak

Mandarin Chinese. About 3.22 x 10 people speak Spanish. In scientific notation, how many more people speak Mandarin Chinese than Spanish?

88

Answer (8.73 x 10 ) – (3.22 x 10 )

(8.73 – 3.22) x 10

5.51 x 10 more people speak Mandarin Chinese than Spanish

88

8

8

Example The Atlantic Ocean has an area of 3.342 x 10

square miles. The Artic Ocean has an area of 5.105 x 10 square miles. In scientific notation, what is the combined area of the two oceans?

7

6