1. Exponents & Scientific Notation
Exponents
An exponent shows how many times a number (the base) is multiplied by itself. For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).
- Base: The number being multiplied (e.g., 2)
- Exponent: How many times to multiply (e.g., 3)
Scientific Notation
Scientific notation expresses very large or very small numbers as a product of a number between 1 and 10 and a power of 10.
Example: \( 4,500,000 = 4.5 \times 10^6 \)
2. Properties & Operations
Order of Operations
The order in which mathematical operations must be performed: PEMDAS:
- Parentheses
- Exponents
- Multiplication & Division (left to right)
- Addition & Subtraction (left to right)
Commutative & Associative Properties
-
Commutative Property: Order doesn't matter (addition/multiplication):
\( a + b = b + a \), \( ab = ba \) -
Associative Property: Grouping doesn't matter (addition/multiplication):
\( (a + b) + c = a + (b + c) \)
Distributive & Identity Properties
- Distributive Property: \( a(b + c) = ab + ac \)
- Identity Property: For addition: \( a + 0 = a \), for multiplication: \( a \times 1 = a \)
Zero Property & Equality Properties
- Zero Property of Multiplication: \( a \times 0 = 0 \)
- Equality Properties: If \( a = b \), then \( a + c = b + c \), \( a - c = b - c \), etc.
3. Factors & Multiples
- Factors: Numbers you can multiply together to get another number. (e.g., factors of 12: 1, 2, 3, 4, 6, 12)
- Multiples: The result of multiplying a number by an integer. (e.g., multiples of 3: 3, 6, 9, 12,...)
4. Variable Expressions & Equations
Understanding Variable Expressions
A variable expression involves numbers, variables (like x), and operations (like +, -, ×, ÷). Examples:
\( 2x + 5 \), \( 7 - y \), \( ab \)
Solving Equations by Addition & Subtraction
To keep equations balanced, do the same operation on both sides.
Example: \( x + 7 = 12 \)
Subtract 7 from both sides: \( x = 5 \)
Solving Equations by Multiplication & Division
Example: \( 3x = 12 \)
Divide both sides by 3: \( x = 4 \)
5. Inequalities & Substitution
Inequalities
- < : Less than
- > : Greater than
- ≤ : Less than or equal to
- ≥ : Greater than or equal to
- ≠ : Not equal to
Example: \( x + 3 > 7 \rightarrow x > 4 \)
Solving Equations & Inequalities by Substitution
Substitute a value for the variable and check if the equation or inequality is true.
Example: In \( 2x = 8 \), substitute \( x = 4 \) to see if \( 2 \times 4 = 8 \).
6. In-Depth Concepts
- Greatest Common Factor (GCF): The largest factor that two numbers share.
- Least Common Multiple (LCM): The smallest multiple that two numbers share.
- Inverse Operations: Operations that undo each other (addition/subtraction, multiplication/division).
- Properties of Equality: Allow you to balance and solve equations.
- Evaluating Expressions: Replacing variables with numbers and performing operations.
7. Examples
- Exponents: \( 5^2 = 25 \)
- Scientific Notation: \( 0.0009 = 9 \times 10^{-4} \)
- Order of Operations: \( 2 + 3 \times 4 = 2 + 12 = 14 \)
- Commutative: \( 4 + 7 = 7 + 4 \)
- Associative: \( (2 + 3) + 4 = 2 + (3 + 4) \)
- Distributive: \( 3(a + 2) = 3a + 6 \)
- Zero Property: \( 9 \times 0 = 0 \)
- Factor: Factors of 15: 1, 3, 5, 15
- Multiple: Multiples of 6: 6, 12, 18, 24,...
- Variable Expression: If \( x = 2 \), then \( 3x + 4 = 10 \)
- Solving Equation: \( x - 5 = 13 \implies x = 18 \)
- Inequality: \( y + 2 < 10 \implies y < 8 \)
- Substitution: \( 2y = 10 \), if \( y = 5 \), \( 2 \times 5 = 10 \)
8. Pre-Test (40 Questions, Multiple Choice)
-
What is \( 3^3 \)?
- A) 6
- B) 9
- C) 27
- D) 12
-
Write 250,000 in scientific notation.
- A) \( 2.5 \times 10^4 \)
- B) \( 2.5 \times 10^5 \)
- C) \( 25 \times 10^4 \)
- D) \( 0.25 \times 10^6 \)
-
What is the value of \( (2 + 4) \times 3 \)?
- A) 18
- B) 14
- C) 12
- D) 20
-
What does the commutative property state?
- A) Changing the grouping doesn’t change the result
- B) Changing the order doesn’t change the result
- C) Multiplying by zero gives zero
- D) Adding zero gives the same number
-
Is \( a(b + c) = ab + ac \) an example of the distributive property?
- A) Yes
- B) No
- C) Only when \( a = 1 \)
- D) Only for addition
-
What is the identity element for addition?
- A) 1
- B) 0
- C) -1
- D) Any number
-
What is \( 7 \times 0 \)?
- A) 0
- B) 7
- C) 1
- D) Undefined
-
If \( a = 7 \), what is \( a + 5 \)?
- A) 10
- B) 12
- C) 2
- D) 35
-
Name two factors of 18.
- A) 2 and 9
- B) 3 and 5
- C) 6 and 11
- D) 1 and 19
-
What is the third multiple of 5?
- A) 5
- B) 10
- C) 15
- D) 20
-
What is the solution to \( x + 8 = 12 \)?
- A) 20
- B) 4
- C) 2
- D) 6
-
Solve: \( 4x = 24 \)
- A) \( x = 6 \)
- B) \( x = 4 \)
- C) \( x = 12 \)
- D) \( x = 8 \)
-
What does \( x < 10 \) mean?
- A) x is greater than 10
- B) x is equal to 10
- C) x is less than 10
- D) x is not equal to 10
-
If \( y = 3 \), what is \( 2y + 1 \)?
- A) 5
- B) 6
- C) 7
- D) 8
-
What is the least common multiple of 4 and 6?
- A) 12
- B) 24
- C) 18
- D) 6
-
What is the greatest common factor of 15 and 25?
- A) 5
- B) 10
- C) 15
- D) 25
-
What is \( 10^0 \)?
- A) 1
- B) 0
- C) 10
- D) 100
-
Write \( 3.2 \times 10^4 \) in standard form.
- A) 32,000
- B) 3,200
- C) 320
- D) 320,000
-
Is \( (a + b) + c = a + (b + c) \) always true?
- A) Yes
- B) No
- C) Only for addition
- D) Only for multiplication
-
What is the solution to \( x - 3 = 7 \)?
- A) 4
- B) 10
- C) 7
- D) 2
-
Give an example of a variable expression.
- A) \( 5 + 4 \)
- B) \( x + 3 \)
- C) \( 7 - 2 \)
- D) \( 12 \times 2 \)
-
If \( x = 6 \), is \( x + 2 = 8 \) true?
- A) Yes
- B) No
- C) Only if \( x = 7 \)
- D) Only if \( x = 8 \)
-
What is \( 5 \times (2 + 3) \)?
- A) 15
- B) 10
- C) 25
- D) 20
-
What is the solution to \( 9y = 81 \)?
- A) 9
- B) 8
- C) 10
- D) 81
-
What is the fourth multiple of 3?
- A) 9
- B) 12
- C) 15
- D) 6
-
What is the solution to \( x/4 = 5 \)?
- A) 9
- B) 20
- C) 1
- D) 4
-
What is the value of \( 2^4 \)?
- A) 8
- B) 16
- C) 12
- D) 24
-
Write 0.004 in scientific notation.
- A) \( 4 \times 10^{-3} \)
- B) \( 4 \times 10^{-2} \)
- C) \( 4 \times 10^{-4} \)
- D) \( 4 \times 10^{3} \)
-
If \( m = 2 \), what is \( 3m \)?
- A) 6
- B) 5
- C) 7
- D) 3
-
Solve: \( y + 9 = 15 \)
- A) 4
- B) 6
- C) 5
- D) 7
-
What is the identity element for multiplication?
- A) 0
- B) 1
- C) -1
- D) 10
-
What is \( 8 \times 1 \)?
- A) 8
- B) 1
- C) 0
- D) 9
-
What is the zero property of multiplication?
- A) Multiplying by 1 gives the same number
- B) Multiplying by 0 gives 0
- C) Adding 0 gives the same number
- D) Subtracting 0 gives 0
-
What is the solution to \( t - 2 = 10 \)?
- A) 8
- B) 12
- C) 10
- D) 2
-
What is the commutative property for multiplication?
- A) \( a \times b = b \times a \)
- B) \( (a \times b) \times c = a \times (b \times c) \)
- C) \( a \times 0 = 0 \)
- D) \( a + b = b + a \)
-
If \( x = 4 \), what is \( x^2 \)?
- A) 8
- B) 16
- C) 12
- D) 24
-
What is \( 6 + 7 \)?
- A) 13
- B) 12
- C) 11
- D) 14
-
What does \( x \ge 5 \) mean?
- A) x is less than 5
- B) x is greater than 5
- C) x is less than or equal to 5
- D) x is greater than or equal to 5
-
If \( z = 9 \), what is \( z/3 \)?
- A) 2
- B) 3
- C) 6
- D) 9
-
What is the greatest common factor of 12 and 18?
- A) 2
- B) 3
- C) 6
- D) 12
9. Questions & Answers
- What is an exponent?
An exponent tells how many times a number (the base) is multiplied by itself. - How do you write 120,000 in scientific notation?
\( 1.2 \times 10^5 \) - What is the distributive property?
Multiplying a number by a sum: \( a(b + c) = ab + ac \). - What is the zero property of multiplication?
Any number multiplied by zero is zero. - What is the least common multiple (LCM)?
The smallest multiple that two or more numbers share. - What is a variable expression?
An expression that contains variables, numbers, and operations. - How do you solve \( x + 5 = 12 \)?
Subtract 5 from both sides: \( x = 7 \). - What does \( x > 8 \) mean?
x is greater than 8. - How do you check your solution in an equation?
Substitute your answer back into the original equation. - What is the identity property of addition?
Any number plus zero is itself.
10. Post-Test (40 Questions)
- What is \( 4^2 \)?
- Write 0.0003 in scientific notation.
- What is the value of \( (3 + 5) \times 2 \)?
- What does the associative property state?
- Is \( ab = ba \) an example of commutative property?
- What is the identity element for multiplication?
- What is \( 12 \times 0 \)?
- If \( a = 5 \), what is \( a - 2 \)?
- Name two multiples of 7.
- What is the fifth multiple of 4?
- What is the solution to \( x + 3 = 11 \)?
- Solve: \( 5x = 35 \)
- What does \( x > 6 \) mean?
- If \( y = 4 \), what is \( 3y - 2 \)?
- What is the least common multiple of 3 and 5?
- What is the greatest common factor of 20 and 30?
- What is \( 2^0 \)?
- Write \( 5.7 \times 10^3 \) in standard form.
- Is \( a + (b + c) = (a + b) + c \) always true?
- What is the solution to \( x - 6 = 9 \)?
- Give another example of a variable expression.
- If \( x = 3 \), is \( x + 7 = 10 \) true?
- What is \( 6 \times (1 + 2) \)?
- What is the solution to \( 7y = 49 \)?
- What is the second multiple of 8?
- What is the solution to \( x/2 = 9 \)?
- What is the value of \( 3^3 \)?
- Write 800,000 in scientific notation.
- If \( m = 7 \), what is \( 2m \)?
- Solve: \( y + 5 = 17 \)
- What is the identity element for addition?
- What is \( 15 \times 1 \)?
- What is the zero property of addition?
- What is the solution to \( t - 8 = 4 \)?
- What is the commutative property for addition?
- If \( x = 5 \), what is \( x^2 \)?
- What is \( 9 + 6 \)?
- What does \( y \leq 12 \) mean?
- If \( z = 12 \), what is \( z/4 \)?
- What is the greatest common factor of 18 and 30?