1 Lesson 1-9 Powers and Laws of Exponents

Location of Exponent An exponent is a little number high and to the right of a regular or base number. An exponent is a little number high and to the right of a regular or base number. 3 4 Base Exponent

Definition of Exponent An exponent tells how many times a number is multiplied by itself. An exponent tells how many times a number is multiplied by itself. 3 4 Base Exponent

What an Exponent Represents An exponent tells how many times a number is multiplied by itself. An exponent tells how many times a number is multiplied by itself. 3 4 = 3 x 3 x 3 x 3

How to read an Exponent This exponent is read three to the fourth power. This exponent is read three to the fourth power. 3 4 Base Exponent

How to read an Exponent This exponent is read three to the 2 nd power or three squared. This exponent is read three to the 2 nd power or three squared. 3 2 Base Exponent

How to read an Exponent This exponent is read three to the 3rd power or three cubed. This exponent is read three to the 3rd power or three cubed. 3 3 Base Exponent

Read These Exponents

What is the Exponent? 2 x 2 x 2 =2 3

What is the Exponent? 3 x 3 =3 2

What is the Exponent? 5 x 5 x 5 x 5 =5 4

What is the Base and the Exponent? 8 x 8 x 8 x 8 =8 4

What is the Base and the Exponent? 7 x 7 x 7 x 7 x 7 =7 5

What is the Base and the Exponent? 9 x 9 =9 2

How to Multiply Out an Exponent to Find the Standard Form = 3 x 3 x 3 x

What is the Base and Exponent in Standard Form? 4 2 = 16

What is the Base and Exponent in Standard Form? 2 3 = 8

3 2 = 9

5 3 = 125

Exponents Are Often Used in Area Problems to Show the Feet Are Squared Length x width = area A pool is a rectangle Length = 30 ft. Width = 15 ft. Area = 30 x 15 = 450 ft. 2 15ft. 30ft

Exponents Are Often Used in Volume Problems to Show the Centimeters Are Cubed Length x width x height = volume A box is a rectangle Length = 10 cm. Width = 10 cm. Height = 20 cm. Volume = 20 x 10 x 10 = 2,000 cm. 3 10

Here Are Some Areas Change Them to Exponents 40 feet squared = 40 ft. 56 sq. inches = 56 in. 38 m. squared = 38 m. 56 sq. cm. = 56 cm

Here Are Some Volumes Change Them to Exponents 30 feet cubed = 30 ft. 26 cu. inches = 26 in. 44 m. cubed = 44 m. 56 cu. cm. = 56 cm

24 Law of Exponents for Multiplication To multiply two powers that have the same base, keep the base and add the exponents. To multiply two powers that have the same base, keep the base and add the exponents. x a x b = x a+b Examples : = = 9 13

25 Law of Exponents for Division To divide two powers that have the same base, keep the base and subtract the exponents. To divide two powers that have the same base, keep the base and subtract the exponents. x a ÷ x b = x a-b Examples : 7 5 ÷ 7 3 = ÷ 2 2 = 2 6 Remember that division can also be written vertically: Now here’s a harder one! 25

26 Law of Exponents for Zero A number to the zero power always equals 1. A number to the zero power always equals 1. Simplify:

27 But what happens if you add or subtract the exponents and you get a negative number ? First of all, there is no crying in math! Second, we have a law for that too! It’s called the Negative Rule! Let me tell you all about it…

28 Negative Rule Any non-zero number raised to a negative power equals its reciprocal raised to the opposite positive power. Any non-zero number raised to a negative power equals its reciprocal raised to the opposite positive power. WHAT!!

29 …Negative Rule Remember that a reciprocal is the multiplicative inverse. In simple terms, flip the fraction! The reciprocal of is. Remember that a reciprocal is the multiplicative inverse. In simple terms, flip the fraction! The reciprocal of is. If we apply the negative rule ( Any non-zero number raised to a negative power equals its reciprocal raised to the opposite positive power ) then, If we apply the negative rule ( Any non-zero number raised to a negative power equals its reciprocal raised to the opposite positive power ) then, In this example, the negative in front of the four remains. Only the negative of the exponent is effected. A non-zero raised to a negative power = The reciprocal raised to the opposite power

30 Power Rule When raising a power to a power, keep the base and multiply the exponents. When raising a power to a power, keep the base and multiply the exponents. (x a ) b = x ab Examples: (2 4 ) 3 = 2 12 (x 3 ) 5 = x 15 Let me jot this down. Oh yes, I got it now!

31 Product to a Power Rule A product raised to a power is equal to each base in the product raised to that exponent. A product raised to a power is equal to each base in the product raised to that exponent. (7 3) 2 = = 49 9 =441 (x 3 y 2 ) 5 = x 15 y 10 (2x 2 yz -3 ) -4 = 2 -4 x -8 y -4 z 12 = = (x y) 2 = x 2 y 2 Here’s one where the variables have exponents Here’s one where the product is raised to a negative power! Examples: Tricky, trickier, trickiest – But I think I got it!

32 Quotient to a Power Rule A quotient raised to a power is equal to each base in the numerator and denominator raised to that exponent. A quotient raised to a power is equal to each base in the numerator and denominator raised to that exponent. Examples: …and this is the last law! 32

33 Why does anything to the zero power equal 1? 2 2 = 2 x 2 = = 2 = = 2 x 2 x 2 = = 2 x 2 x 2 x 2 = = 2 x 2 x 2 x 2 x 2 = 32 Division is a good way of showing how this works: Take the product for 2 5 and divide it by ÷ 2 = 16 and 16 = 2 4 Now take that answer, 16, which is the standard form of 2 4, and divide it by ÷ 2 = 8 and 8= 2 3 Now take that answer, 8, which is the standard form of 2 3, and divide it by 2. 8 ÷ 2 = 4 and 4= 2 2 Now take that answer, 4, which is the standard form of 2 2, and divide it by 2. 4 ÷ 2 = 2 and 2 = 2 1 Now take that answer, 2, which is the standard form of 2 1, and divide it by 2. 2 ÷ 2 = 1 AND 1 = = 1 = Really! THIS WORKS FOR ALL NUMBERS – CLICK HERE TO SEE ONE MORE EXAMPLE! HERE

= 5 X 5= = 5 = = = 5 X 5 X 5= = 5 X 5 X 5 X 5 = = 5 X 5 X 5 X 5 X 5 = 3125 Take the product for 5 5 and divide it by ÷ 5 = 625 and 625 = 5 4 Now take that answer, 625, which is the standard form of 5 4, and divide it by ÷ 5 = 125 and 125 = 5 3 Now take that answer, 125, which is the standard form of 5 3, and divide it by ÷ 5 = 25 and 25 = 5 2 Now take that answer, 25, which is the standard form of 5 2, and divide it by 5 25 ÷ 5 = 5 and 5 = 5 1 Now take that answer, 5, which is the standard form of 5 1, and divide it by 2. 5 ÷ 5 = 1 AND THEREFORE 1 =5 0 Click to go back to where I left off