Laws of exponents
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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

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

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.


Laws of exponents

Exponents

exponent

7

3

base

power


Laws of exponents

  • 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

Squares

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

3 squared =

= 3 x 3 = 9


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

Perfect Squares


Square roots

Square Roots

  • A square root goes the other direction.

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

9

3


Laws of exponents

Square Roots


Radicals

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.

    = 2

4

16


Radicals1

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.


Laws of exponents

Example

  • Simplify:

    a)c)

    b)d)

3

4

27

81

3

8

16


Laws of exponents1

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

Laws of Exponents


Laws of exponents2

Laws of Exponents


Zero exponents

Zero Exponents

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

0

= 1

a


Negative exponents

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.


Laws of exponents

Basic Examples


Basic examples

Basic Examples


Laws of exponents

Basic Examples


Examples

Examples

1.

2.

3.

4.


Scientific notation

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.


Laws of exponents

  • 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.


Laws of exponents

  • 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

Example

  • Write 7,024,000 in scientific notation


Example1

Example

  • Write 476.23 in scientific notation.


Laws of exponents

  • 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.


Laws of exponents

  • 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


Example2

Example

Write 0.0652 in scientific notation.


Example3

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


Answers

Answers

1. a.) 0.0046

b.) 4600000

2. 8.75 x 10

8


Computing with scientific notation

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 10 x 2 x 10

(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


Examples1

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


Answers1

Answers

9

  • 1. 2.484 x 10

  • 2. 4 x 10

  • 3. 1.25 x 10

7

-7


Adding and subtracting

Adding and Subtracting

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


Example4

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


Answer

Answer

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


Example5

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?

6


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