Podcast
Questions and Answers
Which number system is closed under both addition and multiplication?
Which number system is closed under both addition and multiplication?
- Rational Numbers
- Whole Numbers
- Integers
- Natural Numbers (correct)
What is the identity element for addition in the system of whole numbers?
What is the identity element for addition in the system of whole numbers?
- -1
- 1
- 10
- 0 (correct)
Which of the following is an example of an irrational number?
Which of the following is an example of an irrational number?
- $\sqrt{4}$
- -3
- $\pi$ (correct)
- $rac{1}{2}$
Estimating and rounding off numbers are techniques primarily used to:
Estimating and rounding off numbers are techniques primarily used to:
What is the purpose of 'compensating' when estimating calculations?
What is the purpose of 'compensating' when estimating calculations?
What is the definition of a common multiple of two or more numbers?
What is the definition of a common multiple of two or more numbers?
What is the purpose of prime factorization?
What is the purpose of prime factorization?
Which method is used to determine both the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of two numbers?
Which method is used to determine both the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of two numbers?
If Moeneba collects apples at a rate of 5 apples per minute, how many apples does she collect in 15 minutes?
If Moeneba collects apples at a rate of 5 apples per minute, how many apples does she collect in 15 minutes?
In a biscuit recipe, 5 parts of flour are mixed with 2 parts of oatmeal and 1 part of cocoa powder. What is the ratio of flour to oatmeal?
In a biscuit recipe, 5 parts of flour are mixed with 2 parts of oatmeal and 1 part of cocoa powder. What is the ratio of flour to oatmeal?
What formula is used to calculate average speed?
What formula is used to calculate average speed?
What is the difference between simple interest and compound interest?
What is the difference between simple interest and compound interest?
Adding a negative number to a positive number is equivalent to:
Adding a negative number to a positive number is equivalent to:
What is the result of subtracting a negative number from a positive number?
What is the result of subtracting a negative number from a positive number?
According to the distributive property, what is the equivalent expression for $a(b + c)$?
According to the distributive property, what is the equivalent expression for $a(b + c)$?
What is the result of multiplying two negative integers?
What is the result of multiplying two negative integers?
What happens when a number is added to its additive inverse?
What happens when a number is added to its additive inverse?
Which of the following statements is always true when (x) is a real number?
Which of the following statements is always true when (x) is a real number?
How does subtracting a negative number affect the original number?
How does subtracting a negative number affect the original number?
What is the primary reason for the invention of negative numbers?
What is the primary reason for the invention of negative numbers?
If $a$ is a positive number and $b$ is a negative number, which of the following is always negative?
If $a$ is a positive number and $b$ is a negative number, which of the following is always negative?
What is the value of $(-1)^{-1}$?
What is the value of $(-1)^{-1}$?
What does $a^{-n}$ represent?
What does $a^{-n}$ represent?
Based on the laws of exponents, what is the simplified form of $rac{a^5}{a^2}$?
Based on the laws of exponents, what is the simplified form of $rac{a^5}{a^2}$?
A number in scientific notation is written as $3.2 imes 10^{-4}$. What is this number in decimal form?
A number in scientific notation is written as $3.2 imes 10^{-4}$. What is this number in decimal form?
Simplify the expression: $(2^3 imes 2^2) \div 2^4$.
Simplify the expression: $(2^3 imes 2^2) \div 2^4$.
If $a^m = a^n$, what must be true?
If $a^m = a^n$, what must be true?
What is $5^0$?
What is $5^0$?
Evaluate $(-3)^3$.
Evaluate $(-3)^3$.
Given that $x$ is an integer and $x < 0$, which of the following expressions will always yield a positive result?
Given that $x$ is an integer and $x < 0$, which of the following expressions will always yield a positive result?
Solve for $x$: $2^{x+1} = 8$.
Solve for $x$: $2^{x+1} = 8$.
What is the value of $x$ if $5^x = rac{1}{25}$?
What is the value of $x$ if $5^x = rac{1}{25}$?
If the cost price of an item is $C$, and it is sold for $S$, where $S < C$, then there is a:
If the cost price of an item is $C$, and it is sold for $S$, where $S < C$, then there is a:
A journey is completed in two sections. The first section covers 120 km at an average speed of 60 km/h, and the second section covers 180 km at an average speed of 90 km/h. What is the total time for the journey?
A journey is completed in two sections. The first section covers 120 km at an average speed of 60 km/h, and the second section covers 180 km at an average speed of 90 km/h. What is the total time for the journey?
Consider two numbers, $x$ and $y$. If $x$ is a positive integer and $y$ is a negative integer, which of the following expressions results in the largest value?
Consider two numbers, $x$ and $y$. If $x$ is a positive integer and $y$ is a negative integer, which of the following expressions results in the largest value?
A store offers a 20% discount on all items. If a customer buys two items, one priced at $50 and another at $30, what is the total amount the customer will pay after the discount?
A store offers a 20% discount on all items. If a customer buys two items, one priced at $50 and another at $30, what is the total amount the customer will pay after the discount?
Determine which of the following numbers, when raised to any positive integer power, will always result in a number less than or equal to the original number?
Determine which of the following numbers, when raised to any positive integer power, will always result in a number less than or equal to the original number?
Which number system includes both rational and irrational numbers?
Which number system includes both rational and irrational numbers?
What does 'prime factorization' involve?
What does 'prime factorization' involve?
If $a$ and $b$ are integers, which expression always results in an integer?
If $a$ and $b$ are integers, which expression always results in an integer?
What is the result of adding an integer to its additive inverse?
What is the result of adding an integer to its additive inverse?
According to the laws of exponents, what is the simplified form of $(a^3)^4$?
According to the laws of exponents, what is the simplified form of $(a^3)^4$?
What is the value of $(-10)^0$?
What is the value of $(-10)^0$?
What is the value of $x$ in the equation $3^{x} = 81$?
What is the value of $x$ in the equation $3^{x} = 81$?
What is the main difference between simple interest and compound interest?
What is the main difference between simple interest and compound interest?
What is the result of the expression $5 - (-3)$?
What is the result of the expression $5 - (-3)$?
Which expression is equivalent to $x(y - z)$ according to the distributive property?
Which expression is equivalent to $x(y - z)$ according to the distributive property?
What is the sign of the result when a positive integer is divided by a negative integer?
What is the sign of the result when a positive integer is divided by a negative integer?
What is the value of $a^{-3}$?
What is the value of $a^{-3}$?
What is the decimal form of the number $6.2 imes 10^{5}$?
What is the decimal form of the number $6.2 imes 10^{5}$?
If $b^x = b^y$ and $b \neq 0, 1, -1$, what can you conclude about $x$ and $y$?
If $b^x = b^y$ and $b \neq 0, 1, -1$, what can you conclude about $x$ and $y$?
The price of an item is marked up by 25%, but then it is sold at a 10% discount off the marked price. What is the overall percentage profit?
The price of an item is marked up by 25%, but then it is sold at a 10% discount off the marked price. What is the overall percentage profit?
In scientific notation, which of the following represents 0.0000075?
In scientific notation, which of the following represents 0.0000075?
If a store increases the price of an item by 15% and then decreases it by 10%, what is the net percentage change in the price?
If a store increases the price of an item by 15% and then decreases it by 10%, what is the net percentage change in the price?
If $x < 0$, which of the following expressions is always positive?
If $x < 0$, which of the following expressions is always positive?
If the number 0.000047 is expressed in scientific notation as $4.7 \times 10^n$, what is the value of $n$?
If the number 0.000047 is expressed in scientific notation as $4.7 \times 10^n$, what is the value of $n$?
Which of the following numbers is NOT a whole number?
Which of the following numbers is NOT a whole number?
What is the Highest Common Factor (HCF) of 24 and 36?
What is the Highest Common Factor (HCF) of 24 and 36?
If the exchange rate is $1 = 0.85, how many euros would you get for $200, after a commission of 2% is deducted?
If the exchange rate is $1 = 0.85, how many euros would you get for $200, after a commission of 2% is deducted?
If a number $x$ is decreased by 20% and then increased by 25%, what is the net percentage change from the original value?
If a number $x$ is decreased by 20% and then increased by 25%, what is the net percentage change from the original value?
A car travels 240 km in 3 hours. If the car maintains the same average speed, how long will it take to travel an additional 320 km?
A car travels 240 km in 3 hours. If the car maintains the same average speed, how long will it take to travel an additional 320 km?
A retailer buys an item for $80 and marks it up by 40%. If they then offer a discount of 15% on the marked price, what is the final selling price?
A retailer buys an item for $80 and marks it up by 40%. If they then offer a discount of 15% on the marked price, what is the final selling price?
Two cyclists start at the same point and travel in opposite directions. One cyclist travels at 20 km/h and the other at 25 km/h. How far apart are they after 2.5 hours?
Two cyclists start at the same point and travel in opposite directions. One cyclist travels at 20 km/h and the other at 25 km/h. How far apart are they after 2.5 hours?
What is the value of $x$ in the equation $\frac{2^x}{2^3} = 16$?
What is the value of $x$ in the equation $\frac{2^x}{2^3} = 16$?
A jacket is priced at $120. If a customer has a coupon for 15% off and pays with a credit card that offers an additional 5% discount, what is the final price they pay?
A jacket is priced at $120. If a customer has a coupon for 15% off and pays with a credit card that offers an additional 5% discount, what is the final price they pay?
John invests $5,000 in a simple interest account with an annual interest rate of 4%. How much interest will he earn after 3 years?
John invests $5,000 in a simple interest account with an annual interest rate of 4%. How much interest will he earn after 3 years?
Consider the expression $x^2 + 2x + 1$. What is its value when $x = -1$?
Consider the expression $x^2 + 2x + 1$. What is its value when $x = -1$?
If $f(x)=x^2-4x+3$, find $f(2)$.
If $f(x)=x^2-4x+3$, find $f(2)$.
If $f(x) = 3x^3 - 2x + 5$ and $g(x) = x^2 - x + 2$, what is the value of $(f+g)(1)$?
If $f(x) = 3x^3 - 2x + 5$ and $g(x) = x^2 - x + 2$, what is the value of $(f+g)(1)$?
Which set of numbers includes 0 and all counting numbers?
Which set of numbers includes 0 and all counting numbers?
Which of the following number systems is NOT always closed under subtraction?
Which of the following number systems is NOT always closed under subtraction?
Which number, when added to any whole number, leaves the whole number unchanged?
Which number, when added to any whole number, leaves the whole number unchanged?
Which of the following correctly describes a key attribute of integers?
Which of the following correctly describes a key attribute of integers?
Which of the following numbers cannot be expressed as a fraction of two integers?
Which of the following numbers cannot be expressed as a fraction of two integers?
What set encompasses both rational and irrational numbers?
What set encompasses both rational and irrational numbers?
What is the primary goal when estimating calculations?
What is the primary goal when estimating calculations?
Why is 'compensating' important in estimation?
Why is 'compensating' important in estimation?
What are consecutive multiples of a number?
What are consecutive multiples of a number?
What is the Lowest Common Multiple (LCM) of two or more numbers?
What is the Lowest Common Multiple (LCM) of two or more numbers?
How is the Highest Common Factor (HCF) determined using prime factorization?
How is the Highest Common Factor (HCF) determined using prime factorization?
If a mixture requires a ratio of 3 parts sand to 1 part cement, and you have 6 parts of sand, how much cement do you need?
If a mixture requires a ratio of 3 parts sand to 1 part cement, and you have 6 parts of sand, how much cement do you need?
A car travels 200 km in 4 hours. What formula would you use to find its average speed?
A car travels 200 km in 4 hours. What formula would you use to find its average speed?
What financial concept involves paying a deposit followed by monthly installments?
What financial concept involves paying a deposit followed by monthly installments?
Adding a negative number to a positive number is equivalent to which operation?
Adding a negative number to a positive number is equivalent to which operation?
What is the result of multiplying a positive integer by a negative integer?
What is the result of multiplying a positive integer by a negative integer?
If a number is added to its additive inverse, what is the result?
If a number is added to its additive inverse, what is the result?
What is the result of dividing a negative number by another negative number?
What is the result of dividing a negative number by another negative number?
Express the number 56,700 in scientific notation.
Express the number 56,700 in scientific notation.
Solve for $x$: $3^{x} = 27$.
Solve for $x$: $3^{x} = 27$.
Simplify: $\frac{5^6 \times 5^{-2}}{5^2}$
Simplify: $\frac{5^6 \times 5^{-2}}{5^2}$
If $p$ and $q$ are integers such that $p < 0$ and $q > 0$, which expression is always negative?
If $p$ and $q$ are integers such that $p < 0$ and $q > 0$, which expression is always negative?
What is the result of subtracting a negative number from another negative number?
What is the result of subtracting a negative number from another negative number?
If $x$ is a real number, which of the following is always non-negative?
If $x$ is a real number, which of the following is always non-negative?
Which expression is equivalent to $-(a - b)$?
Which expression is equivalent to $-(a - b)$?
Given the expression $(-1)^n$, under what condition will the result always be -1?
Given the expression $(-1)^n$, under what condition will the result always be -1?
If $f(x) = x^2 - 3x + 2$, for what values of $x$ does $f(x) = 0$?
If $f(x) = x^2 - 3x + 2$, for what values of $x$ does $f(x) = 0$?
Let $a, b, c$ be integers such that $a + b + c = 0$. What is the minimum possible value of $a^2 + b^2 + c^2$ if at least one of $a, b, c$ is nonzero?
Let $a, b, c$ be integers such that $a + b + c = 0$. What is the minimum possible value of $a^2 + b^2 + c^2$ if at least one of $a, b, c$ is nonzero?
Flashcards
Natural Numbers
Natural Numbers
Numbers used for counting, starting from 1: 1, 2, 3,...
Whole Numbers
Whole Numbers
Natural numbers plus zero: 0, 1, 2, 3,...
Integers
Integers
Extends whole numbers to include negative numbers.
Rational Numbers
Rational Numbers
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Irrational Numbers
Irrational Numbers
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Real Numbers
Real Numbers
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Estimating
Estimating
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Rounding Off
Rounding Off
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Compensating for Errors
Compensating for Errors
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Consecutive Multiples
Consecutive Multiples
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Common Multiple
Common Multiple
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Lowest Common Multiple (LCM)
Lowest Common Multiple (LCM)
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Prime Factorization
Prime Factorization
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Highest Common Factor (HCF)
Highest Common Factor (HCF)
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Ratio
Ratio
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Rate
Rate
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Proportion
Proportion
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Discount
Discount
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Profit
Profit
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Loss
Loss
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Cost Price
Cost Price
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Marked Price
Marked Price
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Selling Price
Selling Price
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Hire Purchase
Hire Purchase
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Simple Interest
Simple Interest
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Compound Interest
Compound Interest
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Currency Exchange
Currency Exchange
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Commission
Commission
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Adding a Negative
Adding a Negative
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Subtracting a Negative
Subtracting a Negative
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Additive Inverse
Additive Inverse
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Exponents
Exponents
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Exponent Law: Multiplication
Exponent Law: Multiplication
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Exponent Law: Division
Exponent Law: Division
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Exponent Law: Power of a Power
Exponent Law: Power of a Power
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Whole Number System
Whole Number System
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Identity Element for Addition
Identity Element for Addition
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Compensating
Compensating
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Average Speed Formula
Average Speed Formula
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Hire Purchase Interest
Hire Purchase Interest
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Pos x Neg = ?
Pos x Neg = ?
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Neg x Neg = ?
Neg x Neg = ?
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Distributive Property
Distributive Property
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Pos ÷ Neg = ?
Pos ÷ Neg = ?
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Neg ÷ Neg = ?
Neg ÷ Neg = ?
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Negative Numbers
Negative Numbers
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Scientific Notation
Scientific Notation
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Scientific Notation (Large Numbers)
Scientific Notation (Large Numbers)
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Scientific Notation (Small Numbers)
Scientific Notation (Small Numbers)
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Decimal Form Conversion
Decimal Form Conversion
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Counting Numbers
Counting Numbers
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Simplifying Exponential Expressions
Simplifying Exponential Expressions
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Solving Exponential Expressions
Solving Exponential Expressions
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Word Problems Involving Integers
Word Problems Involving Integers
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Closure Property
Closure Property
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The Purpose of Integers
The Purpose of Integers
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Compensating for Rounding
Compensating for Rounding
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Total HP Price
Total HP Price
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Exponent Law: Product to a Power
Exponent Law: Product to a Power
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Square of a Number
Square of a Number
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Cube of a Number
Cube of a Number
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Square Root
Square Root
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Anything to the power 0.
Anything to the power 0.
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Study Notes
Properties of Numbers
- Natural numbers are used for counting and begin with 1.
- Adding or multiplying natural numbers results in another natural number.
- Subtracting or dividing natural numbers does not always result in a natural number.
- Whole numbers include natural numbers and 0.
- Adding 0 to a number does not change the number (Identity Element for Addition).
- Integers extend whole numbers to include negative numbers.
- The sum of two integers can be zero, and each integer has an additive inverse.
- Rational numbers can be expressed as a quotient of two integers.
- Irrational numbers cannot be expressed as a fraction of two integers (e.g., (\sqrt{2}), (\sqrt{5}), (\pi)).
- Real numbers include both rational and irrational numbers.
Calculations with Whole Numbers
- Estimating involves approximating answers without precise calculations.
- Rounding off simplifies calculations.
- Compensation corrects errors introduced by rounding.
- Adding, multiplying, and subtracting can be performed using column methods.
- Long division involves dividing numbers step-by-step.
Multiples and Factors
- Consecutive multiples are obtained by multiplying a number by 1, 2, 3, etc.
- A common multiple is a number that is a multiple of two or more numbers.
- The Lowest Common Multiple (LCM) is the smallest common multiple of two or more numbers.
- Prime factorization expresses a number as a product of its prime factors.
- The Highest Common Factor (HCF) is the largest number that divides each number without a remainder.
- To find the LCM, multiply all prime factors of both numbers, without repeating.
- To find the HCF, multiply the common prime factors of the numbers.
Solving Problems about Ratio, Rate, and Proportion
- Rate example: Moeneba collects about 5 apples per minute.
- Ratios compare collection rates between different people (e.g., Garth collects about 12, and Kate collects about 15 apples per minute).
- Biscuit recipe example: 5 parts flour, 2 parts oatmeal, 1 part cocoa powder.
- Average speed = Distance / Time.
- Distance = Average speed × Time.
- Time = Distance / Average speed.
Solving Problems in Financial Contexts
- Percentage calculations determine discounts, profits, and losses.
- Key terms: cost price, marked price, selling price.
- Hire purchase involves a deposit and monthly instalments.
- Total HP price is the deposit plus the total instalments.
- Interest is the difference between the HP price and the cash price.
- Simple interest is calculated on the principal amount.
- Compound interest adds interest to the principal each year.
- Currency exchange converts one currency to another based on the exchange rate, with potential commissions.
Adding and Subtracting with Integers
- Adding a negative number is equivalent to subtracting the corresponding positive number.
- Subtracting a negative number is equivalent to adding the corresponding positive number.
- Subtracting a larger number from a smaller number results in a negative number.
- Terms:
- ( a + (-b) = a - b )
- ( a - (-b) = a + b )
- ( a + (-a) = 0 )
- ( a - (-a) = a + a )
Multiplying and Dividing with Integers
- The product of two positive numbers is positive.
- The product of a positive number and a negative number is negative.
- The product of two negative numbers is positive.
- Distributive Property: ( a(b + c) = ab + ac ).
- The quotient of a positive number and a negative number is negative.
- The quotient of two negative numbers is positive.
- Adding an integer has the same effect as subtracting its additive inverse.
- Subtracting an integer has the same effect as adding its additive inverse.
Powers, Roots, and Word Problems
- The square of ( x ) is ( x^2 ).
- The cube of ( x ) is ( x^3 ).
- The square root of ( x ) is ( \sqrt{x} ).
- Positive and negative numbers can have square and cube roots.
- Integers are used in real-world problems such as temperature changes and financial transactions.
Integers
- Negative numbers (-7, -500, -3/4, -3.46) are additive inverses of whole numbers.
- When a larger number is subtracted from a smaller one, the result is negative (e.g., 5 − 12 = −7).
- Properties of Negative Numbers:
- Adding a Negative Number: Equivalent to subtracting the corresponding positive number.
- Subtracting a Negative Number: Equivalent to adding the corresponding positive number.
- Product of a Positive and a Negative Number: The result is a negative number.
Exponents
- Exponents indicate repeated multiplication (e.g., (5 \times 5 \times 5 = 5^3)).
- In mixed operations, powers are calculated before multiplication and division.
- Laws of Exponents:
- (a^m \times a^n = a^{m+n})
- (a^m \div a^n = a^{m-n})
- ((a^m)^n = a^{m \times n})
- ((a \times b)^n = a^n \times b^n)
- (a^0 = 1)
Order of Operations
- Parentheses/Brackets
- Exponents/Orders
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
- Negative exponents: (a^{-n} = \frac{1}{a^n}).
Scientific Notation
- Expresses numbers as ( \pm a \times 10^n), where (1 \leq a < 10) and (n) is an integer.
- Large numbers: Move the decimal point to the left; (n) is positive.
- Small numbers: Move the decimal point to the right; (n) is negative.
- Simplifying Exponential Expressions: Use the laws of exponents.
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