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# Laws of Exponents - PowerPoint PPT Presentation

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.

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Laws of Exponents

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## Laws of Exponents

Objective:

TSW simplify powers.

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

exponent

7

3

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 squared =

= 3 x 3 = 9

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

9

3

Square Roots

- 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.

= 2

4

16

• 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)

3

4

27

81

3

8

16

### 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

### Zero Exponents

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

0

= 1

a

### Negative Exponents

)

(

1

-n

a

______

=

n

a

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

Basic Examples

Basic Examples

1.

2.

3.

4.

## Scientific Notation

Objective:

TSW rewrite numbers in scientific notation

TSW 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?

6

1. a.) 0.0046

b.) 4600000

2. 8.75 x 10

8

### 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

5

• 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)

-4

-4

9

• 1. 2.484 x 10

• 2. 4 x 10

• 3. 1.25 x 10

7

-7

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

### Example

8

• 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?

8

8

8

• (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

8

8

### Example

7

• 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?

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