1st Semester Review

Laws of Exponents

a^-n is the reciprocal of aⁿ. For example, 4² and

4^-2 are inverses or reciprocals.

4²  = 4(4) = 16   4^-2 = 1 / 4(4) =1 / 16

If this is the case, if 3³ = 27, then 3^-3

Multiple Choice

If  $$3^{3\ }=\ 27,\ then\ 3\ ^{-3\ }=\ ?$$  

• $$\frac{1}{3^3}$$  

• -3

• -27

• $$\frac{1}{3^{-3}}$$  

Negative Exponents...

Multiple Choice

$$If\ 2^{-4}=\ \frac{1}{2^4},\ then\ 5^{-2\ }=\ ?$$   

• -25

• $$\frac{1}{5^2}$$  

• -5

• $$\frac{1}{5^{-2}}$$  

Multiplying Exponents

When multiplying exponents, you simply add the exponents.

2³(2⁵) expanded out is 2(2)(2) times 2(2)(2)(2)(2)

You keep the base, 2, and add the exponents 3 + 5 = 8. 2³⁺⁵ = 2⁸

This rule applies to variables as well.

Multiple Choice

Simplify a⁴(a⁵).

• a⁴⁺⁵ = a⁹

• a^4-5 = a^-1

• a⁴⁽⁵⁾ = a²⁰

• 2a⁴⁺⁵ = 2a⁹

Dividing Exponents

Multiple Choice

$$\frac{12^6}{12^4}=\ ?$$  

• 12

• 1

• $$12^{\left(6+4\right)}\ =\ 12^{10}$$  

• $$12^{\left(6\ -\ 4\right)}\ =\ 12^2$$  

Multiple Choice

Simplify

$$\left(\frac{2^0}{3^2}\right)^{ }$$  

• $$\frac{1}{9}$$  

• 1

• 49

• $$\frac{6}{1}$$  

Some text here about the topic of discussion

​What would you do for this problem?

​Solution:

(n²)³ = n² ⋅ n² ⋅ n²

= n ⋅ n ⋅ n ​⋅ n ⋅ n⋅ n

= n⁶

​

Shortcut: multiply the exponents​

Power of a Power Rule

Multiple Choice

$$\left(10^3\right)^5=$$  

• $$10^{15}$$  

• $$10^8$$  

• $$10^5$$  

• $$10^3$$  

Multiple Choice

$$\left(10^7\right)^7=$$  

• $$10^1$$  

• $$10^7$$  

• $$10^{14}$$  

• $$10^{49}$$  

Algebraic Terminology

Multiple Choice

What are the terms in the expression?

-4x + 6y - 2

• -4x, 6y, -2

• x, y

• 4x, 6y

• -2

Multiple Choice

What are the variables in the expression?

-4x + 6y - 2

• -4x, 6y, -2

• x, y

• 4x, 6y

• -2

Multiple Choice

What are the coefficients in the expression?

-4x + 6y - 2

• -4x, 6y, -2

• x, y

• 4x, 6y

• -4, 6

Multiple Choice

What is the constant in the expression?

-4x + 6y - 2

• -4x, 6y, -2

• x, y

• 4x, 6y

• -2

Multiple Choice

Which of the following expressions contains the coefficients 4 and -6?

• 4n - 6n + 4
• 4n + 6n - 4
• -4n - 6n
• -4n + 6n

Multiple Choice

A(n) __________________ is one or more terms separated by addition or subtraction.

• variable

• term

• coefficient

• expression

Multiple Choice

5y + 2y = 28

• Term

• Expression

• Equation

Multiple Choice

(10- u) = (x + 4)

• Term
• Expression
• Equation

Combining Like Terms!

How many bottles of water do you have?

How many burgers did you order?

w= water

b=burger

d=doritos

Therefore we have 3w+2b+1d

Multiple Choice

Simplify by combining like terms:

$$7b-3b+4f$$  

• 
  

  $$10b+4f$$  

• $$8b$$  

• $$4b-4f$$
  

• $$4b+4f$$   

Multiple Choice

Simplify by combining like terms:

$$4t-t+3$$  

• $$4t-4$$  

• $$5t+3$$  

• $$3t+3$$  

• $$5t-3$$  

Multiple Choice

Simplify by combining like terms:

$$7f+8g+4f+3f+6f+2g$$  

• $$20f+11g$$  

• $$21f+10g$$  

• $$20f+10g$$  

• $$22f+10g$$  

Multiple Choice

Like terms have the same _____ and the same _____.

• variables and numbers
• signs and numbers
• variables and exponents
• variables and signs

Multiple Choice

Simplify the expression:

• 2x²+13y+5

• 2x²+13y-5

• x²-y-5

• 4x²-13y-5

Multiple Choice

Simplify the expression:

• 2x²-2y-2

• -3x²-2y-2

• -2x²-4y-2

• -2x²-2y+2

Distributive Property

Multiple Choice

Simplify the expression
11(9 + d)

• 99 + d

• 20 + 9d

• 20 + d

• 99+11d

Multiple Choice

3(2c - 5)

• 6c - 15
• 6c + 15
• 6c + 15
• -6c - 15

Multiple Choice

2(c - 8)

• 2c - 16
• 2c + 16
• 2c - 8
• c - 16

Multiple Choice

Simplify the expression: 3(1 - 2x) + 7

• 21 - 6x
• 24 + 6x
• 10 + 6x
• 10 - 6x

Multiple Choice

-9(2g - 3)

• 18g + 27
• -18g + 27
• -18g - 27
• 18g - 27

Multiplying Binomials

EQ: How do we use area model to multiply polynomials

I can multiply using area model

Multiple Choice

What number goes in the box with the question mark?

• 3,600

• 3,000

• 3,200

Multiplying with variables

******Please get paper for next problems

Multiple Choice

Write the expanded (general) form of the product (x+3)(x+6).

• x²+18x+9

• x²+9x+18

• x²+18x+18

Multiple Choice

Find the product:
(y - 3)(y + 7)

• y² - 21
• y² + 10x - 21
• y² + 4y - 21
• 2y - 10

Multiple Choice

Simplify
(p − 8)²

 Hint: (p − 8)² = (p − 8)(p − 8)

• p ² − 16p − 64 
• p ² + 16p + 64 
• p ² − 16p − 8
• p ² − 16p + 64 

Multiple Choice

(x + 5)²

• x² + 25

• x² + 10x + 25

• x² + 10 x + 10

• x² + 10

Multiple Choice

 Multiply:
(r + 7)(r − 7) 

• r ² − 49 
• r ² + 14
• r ² − 7r + 49 

Multiple Choice

The length of a rectangle is x + 2 units and the width is 3x + 7

units. Find the area.

• 3x² + 9x + 14 square units

• 3x² + 13x + 14 square units

• 4x + 9 square units

• 8x + 18 square units

Multiple Choice

Find the Product.
(2x + 3)(2x - 3)

• 4x² - 9 

• 4x² + 9

• 4x²- 12x - 9

• 4x² + 12x - 9

Algebraic Translation

Translate the following expressions into algebraic notation:

Multiple Choice

Which algebraic expression best represents the following verbal expression?  

9 times a number, minus 2

• 2 - 9n

• 9  - 2n

• 2n - 9

• 9n - 2

Multiple Choice

Which algebraic expression best represents the following verbal expression?
 
two more than three times a number

• 3n - 2
• 3n + 2
• 3 + 2n
• 3 - 2n

Multiple Choice

Which algebraic expression best represents the following verbal expression?
 
two increased by four times a number is eight

• 2 + 4x = 8
• 2 + 4x + 8
• (2+4)8
• 2(4)x = 8

Multiple Choice

Which algebraic expression best represents the following verbal expression?

three more than the product of 6 and a number is 15

• 3 + 6m + 15

• 3 + 6 + m = 15

• 6m + 3 + 15

• 6m + 3 = 15

Multiple Choice

Which algebraic expression best represents the following verbal expression?

five less than a number squared

• 5 - 2a

• a² - 5

• 5 - a²

• 2a - 5

Multiple Choice

Which algebraic expression best represents the following verbal expression?
 
the sum of a number and four, divided by three is 2

• n + 4 = 2 ÷ 3
• (n + 4) ÷ 3 = 2
• n + (4 ÷ 3) = 2
• n + 4 ÷ 3 + 2

Multiple Choice

5 less than the product of 7 and a number

• 7s - 5

• 5 - 7 + s

• 7(s - 5)

• 5 - 7s

Solving Linear Equations

PEMDAS

• Parenthesis (This means distribute and solve () first)

• Exponents (this includes square roots)

• Multiply or Divide next from left to right

• Add or Subtract next from left to right

Multiple Choice

Simplify:

63 ÷ 7 - 3 ⋅ 2 + 4

• 7

• -9

• -1

• 3

Multiple Choice

Simplify:

24 ÷ 4 ⋅ 2 + (3² - 1 + 6)

• 5

• 14

• 17

• 26

Opposite Operations

• Use Op Op when you don't know the variable.

• SADMEP

• Start furthest away from the variable.

• Get the variable on one side and numbers on the other.

Multiple Choice

What is the opposite operation of addition?

• subtraction

• multiplication

• division

• square root

Multiple Choice

What is the opposite operation of multiplication?

• addition

• subtraction

• division

• square root

Multiple Choice

What is the opposite operation of squaring a number?

• taking the square root

• dividing

• multiplying

• subtracting

Multiple Choice

How do you solve 9x - 4 = 23

• divide both sides by 4, then add 9

• add 4 to both sides, then divide both sides by 9

• divide both sides by 9, then add 4

• add 4 to both sides, then divide by 23

How to Solve for a Variable

• Check that the expression is simplified

• Distribute first

• Combine Like Terms

• Do Opposite Operations to get the Variable alone

Multiple Choice

The first step to solving the equation g - 3 = -12 is to . . .

• divide by -3 on both sides of the equal sign

• multiply by -3 on both sides of the equal sign

• subtract 3 on both sides of the equal sign

• add 3 on both sides of the equal sign

Multiple Choice

What is the first step in solving for w in the equation 3w + 5 = 14?

• Add 5 to both sides of the equal sign

• Divide by 3 on both sides of the equal sign

• Subtract 5 from both sides of the equal sign

• Multiply by 3 on both sides of the equal sign

Multiple Choice

19  + a  =  -50

• a = -950
• a =  -69
• a = -31
• a = -50/19

Multiple Choice

Solve the equation:
-5x = 25

• x = 20
• x = -20
• x = -5
• x = 5

Multiple Choice

3x - 5 = -26

• x=-21

• x=21

• x=-7

• x=61

Multiple Choice

Solve for x: 
4x + 1 = 7x - 5

• -1
• 2
• 6/11
• 8

Multiple Choice

Solve: -7(z - 6) = -70

• 16

• -16

• 8

• -1/2

Word Problems

Use algebra to find the answers to the following word problems:

Multiple Choice

Aliyah had $24 to spend on seven pencils. After buying them she had $10. How much did each pencil cost?

• $1

• $2

• $3

• $4

Multiple Choice

331 students went on a field trip. Six buses were filled and 7 students traveled in cars. How many students were in each bus?

• 56

• 47

• 54

• 26

Multiple Choice

You bought a magazine for $5 and four erasers. You spent a total of $25. How much did each eraser cost?

• $4

• $6

• $3

• $5

Multiple Choice

The sum of three consecutive even numbers is 48. What are the smallest of these numbers?

• 16

• 18

• 12

• 14

Multiple Choice

A father is now four times as old as his son. If the sum of their ages 10 years ago was 60, find the present age of the father.

• 48

• 56

• 60

• 64