Questions and Answers
What does prime factorization help with?
What is the value of a digit based on its position within a number called?
place value
What is the condensed form of expanded form?
standard form
How can numbers be represented to show the place value of each digit?
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What is the method of writing or displaying numbers in terms of a decimal number between 1 and 10 multiplied by a power of 10 called?
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What acronym is used for the order of operations?
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What is an arrangement of objects in equal rows called?
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What law states that any finite sum is unaltered by reordering its terms?
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What law states that any finite product is unaltered by reordering its terms?
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What is the identity property of 1?
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What is the identity property of 0?
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What is the inverse property of addition?
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What does multiplying a number by its multiplicative inverse equal?
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What is the inverse of multiplication?
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What is the inverse operation of addition?
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What is the inverse operation of division?
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What is the associative property of multiplication?
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What is the associative property of addition?
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What is the distributive law of multiplication over addition?
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What is the Greatest Common Factor (GCF)?
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What is a prime factor?
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What is a prime number?
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What are factors?
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What is the result of multiplying two or more factors?
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Study Notes
Prime Factorization and Distributive Property
- Use prime factorization to simplify expressions using the distributive property.
- Steps involve identifying prime factors, multiplying common factors, and simplifying the expression.
Place Value
- Place value indicates the value of a digit based on its position in a number.
- Example: In 356, the digits represent 3 hundreds, 5 tens, and 6 ones.
Number Forms
- Standard Form: The condensed version of a number. Example: 536.
- Expanded Form: A representation that shows individual place values. Example: 536 = 500 + 30 + 6.
Scientific Notation
- A way to express numbers as a decimal between 1 and 10 multiplied by a power of 10.
- Example: 34 can be written as 3.4 × 10^1.
Order of Operations (PEMDAS)
- Follow the sequence: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction; evaluate from left to right.
Arrays
- An arrangement of objects in equal rows, useful for visualizing multiplication.
Commutative Laws
- Addition: The sum remains the same when addends are reordered. Example: a + b = b + a.
- Multiplication: The product is unchanged when factors are reordered. Example: a × b = b × a.
Identity Properties
- Identity Property of 1: Multiplying any number by one yields the original number. Example: 7 × 1 = 7.
- Identity Property of 0: Adding zero to any number leaves it unchanged. Example: 20 + 0 = 20.
Inverse Properties
- Addition: The inverse of a number is found by adding its negative. Example: 5 + (-5) = 0.
- Multiplication: A number multiplied by its inverse equals one. Example: 5 × (1/5) = 1.
- Reciprocal of Multiplication: Two numbers with a product of one are reciprocals.
Inverse Operations
- Multiplication: The inverse operation is division; for example, the inverse of 5 is (1/5).
- Division: The inverse operation is multiplication; e.g., the inverse of (1/5) is × 5.
- Addition: The inverse operation is subtraction; e.g., the inverse of 5 is -5.
- Subtraction: The inverse operation is addition; e.g., the inverse of -5 is 5.
Associative Properties
- Multiplication: Changing the grouping of factors does not affect the product. Example: (2 × 3)4 = (2 × 4)3.
- Addition: Reorganizing groupings of addends does not alter the sum. Example: (2 + 3) + 4 = 2 + (3 + 4).
Distributive Laws
- Distributive Law of Multiplication over Addition: Distributes multiplication across addition. Example: 3(2 + 6) = 3(2) + 3(6).
- Distributive Law of Multiplication over Subtraction: Distributes multiplication across subtraction. Example: 3(2 - 6) = 3(2) - 3(6).
Greatest Common Factor (GCF)
- The largest shared factor of two or more numbers. Example: GCF of 8 and 4 is 4.
- Finding GCF involves identifying prime factors, multiplying common factors, and writing the final result.
Factor Definitions
- Prime Factor: A factor that is a prime number.
- Prime Number: A whole number greater than 1 with exactly two distinct factors: 1 and itself.
- Factors: Numbers multiplied together to yield a product.
Key Arithmetic Components
- Product: The result from multiplying two or more factors.
- Addend: A number that contributes to a sum.
- Sum: The result of adding two or more addends.
- Division Components: Dividend (the number being divided), Divisor (the number that divides) leading to a Quotient (the result of division).
- Subtraction Components: Minuend (the number from which another is subtracted), Subtrahend (the number that is to be subtracted), leading to a Difference (the result of the subtraction).
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