Numerical Expressions and Exponents

​Introduction to Exponents

​Essential Questions

How can I apply the properties of integer exponents to generate equivalent numerical expressions?

How do I simplify and evaluate numeric expressions involving integer exponents?

​Essential Vocabulary for

Exponent -

Base -

Power-​

Exponents are repeated multiplication:

3² = 3 x 3

3³ = 3 x 3 x 3

3^{4 = 3 x 3 x 3 x 3}

​What does an exponent mean?

​The exponent tells us how many times to multiply the base by itself.

​Exponential Form

• ​Exponential Form is a number made up of two pieces:

  • ​The base is the big, bottom number.

  • ​The exponent is the little number in the top right.

Multiple Choice

What does 4³ mean?

• 4 x 4 x 4

• 3 x 3 x 3 x 3

• 4 + 4 + 4

Multiple Choice

Which is equivalent to 5²

• 2 x 2 x 2 x 2 x 2

• 5 + 2

• 5 x 5

Let's evaluate, or solve, some exponents.

4² = 4 x 4

4 x 4 = 16

So, 4² = 16

Multiple Choice

Evaluate: 3²

• 3 + 3 = 6

• 3 x 2 = 6

• 3 x 3 = 9

Multiple Choice

Evaluate:

5³

• 5 x 5 x 5 = 125

• 5 x 3 = 15

• 3 x 3 x 3 x 3 x 3 = 243

Remember: 2⁴ =

2 x 2 x 2 x 2

2 x 2 = 4

4 x 2 = 8

8 x 2 = 16

Multiple Choice

Evaluate: 6³

• 18

• 36

• 216

Multiple Choice

What is the base?

$$5^2=25$$  

• 5

• 2

• 25

Multiple Choice

What is the exponent?

$$2^3=8$$  

• 2

• 3

• 8

Multiple Choice

What is the base?

$$x^4$$  

• x

• 4

• There is no base.

​10s Trick!!

When the base is 10, look at ​exponent, and that is how many zeros are in the answer!

Multiple Choice

How many zeros will be in the answer to  $$10^6$$  ?

• 10

• 6

• 16

• 60

• You can't tell.

Multiple Choice

Solve:  $$10^4$$  

• 10,000,000,000

• $$10,000$$  

• 40

• 1,000

• 100,000

Properties of Exponents

Learn and practice

Multiply Properties

• Product of Powers

• Power of a Power

• Power of a Product

Product of Powers

Words: when multiplying like bases, add the exponents

Symbols: (x^a)(x^b) = x^a+b

Numbers: (8³)(8⁶) = 8⁹

Power of a Power

Words: when a power (base with exponent) is raised to another power (exponent), you multiply the exponents (keep the base!)

Symbols: (x^a)^b= x^ab

Numbers: (8³)⁶=8¹⁸

Poll

My level of understanding

(left to right, NO: I don't get it at all, Whoops: need some help but I'm starting to understand, Nailed It: I can do this on my own.

Multiple Choice

(2m³)(3m⁴) =

• 5m⁷

• 6m¹²

• 5m¹²

• 6m⁷

Division Properties

• Quotient of Powers

• Power of a Quotient

• Zero Exponent

Quotient of Powers

Power of a Quotient

Zero Exponent

Multiple Choice

Simplify:

$$\frac{5^7}{5^4}$$  

• $$5^{11}$$  

• $$1^3$$  

• $$5^3$$  

Multiple Choice

What is 100⁰?

• 0

• 1

• 100

Multiple Choice

What is m⁴ x m⁶ ?

• m⁴⁶

• m²⁴

• m¹⁰

• Unable to solve

Multiple Choice

What is 

$$\frac{5^{15}}{5^5}$$  ?

• $$5^{20}$$  

• $$5^{10}$$  

• $$5^{15}$$  

• $$5^{155}$$  

Power Rule

• To raise a power to a power, multiply the powers

• Ex. 1: (4³)²=

• Ex. 2: (3⁵)⁶=

• Ex. 3: (p⁴)³ =

Multiple Choice

What is (y⁴)⁵

• y²⁰

• y⁵

• y⁹

• y¹

Multiple Choice

Using the product rule, which is equivalent to

$$4^5\times4^3$$  ?

• $$4^8$$  

• $$4^{15}$$  

• $$4^6$$  

• $$4^2$$  

Multiple Choice

Using the product rule, which is equivalent to

$$y^4\times y^3$$  ?

• $$y^{12}$$   

• $$y^1$$  
  

• $$y^7$$  

  
  

Multiple Choice

Using the quotient rule, which is equivalent to

$$6^3\div6^2$$  ?

• $$6^5$$  

• $$6^6$$  

• $$6^1$$  

• $$6^2$$  

Multiple Choice

Using the quotient rule, which is equivalent to

$$\frac{y^{10}}{y^3}$$  ?

• $$y^7$$  

• $$y^2$$  

• $$y^{13}$$  

• $$y^6$$  

Multiple Choice

Using the power rule, which is equivalent to

$$\left(3^4\right)^5$$  ?

• $$3^9$$  

• $$3^{20}$$  

• $$3^1$$  

• $$3^{18}$$  

Multiple Choice

Using the power rule, which is equivalent to

$$\left(x^2\right)^{10}$$  ?

• $$x^8$$  

• $$x^{22}$$  

• $$x^{12}$$  

• $$x^{20}$$  

Multiple Choice

What is the value of 7⁰

• 7

• 1

• 5

• 70

Multiple Choice

What is the exponent?

$$2^3=8$$  

• 2

• 3

• 8

Multiple Choice

What is the base?

$$5^2=25$$  

• 5

• 2

• 25

Multiple Choice

What is the base?

$$x^4$$  

• x

• 4

• There is no base.

Multiple Choice

Write the following in expanded form:

$$5^2\ =$$  

• $$5\times2$$  

• $$5\times5$$  

• $$2\times2\times2\times2\times2$$  

• $$5+5$$  

Multiple Choice

Write the following in expanded form:

$$8^4\ =$$  

• $$8\times4$$  

• $$8\times8\times8\times8$$  

• $$4\times4\times4\times4\times4\times4\times4\times4$$  

• $$8+8+8+8$$  

Multiple Choice

Write the following in expanded form:

$$3^5\ =$$  

• $$3\cdot5$$  

• $$3\cdot3\cdot3\cdot3\cdot3$$  

• $$5\cdot5\cdot5$$  

• $$3+3+3+3+3$$  

Multiple Choice

Write the following in expanded form:

$$x^3=$$  

• $$x\cdot3$$  

• $$x\cdot x\cdot x$$  

• $$x+3$$  

• $$x+x+x$$  

Multiple Choice

Write the following in exponential form:

$$4\times4\times4\times4\times4=$$  

• $$4\times5$$  

• $$4^5$$  

• $$5^4$$  

Multiple Choice

Write the following in exponential form:

$$11\cdot11\cdot11=$$  

• $$11\times3$$  

• $$11^3$$  

• $$3^{11}$$  

Multiple Choice

Write the following in exponential form:

$$a\cdot a\cdot a=$$  

• $$a\times3$$  

• $$a^3$$  

• $$3^a$$  

Multiple Choice

$$4^2\cdot4^6$$  

• $$4^{2\cdot6\ }=\ 4^{12}$$  

• $$4^{2+6\ }=\ 4^8$$  

Multiple Choice

$$\left(3d\right)^3$$  

• $$3d^3$$  

• $$9d^3$$