Podcast
Questions and Answers
Which of the following numbers is an irrational number?
Which of the following numbers is an irrational number?
- $\sqrt{9}$
- $\sqrt{3}$ (correct)
- 0.333...
- $ rac{1}{3}$
Which set of numbers includes only integers?
Which set of numbers includes only integers?
- {-6, -3, 0, $\frac{1}{2}$, 5}
- {-4, -1, 0, 2, $\sqrt{9}$}
- {-5, -2, 0, 4, 7} (correct)
- {-3, -1.5, 0, 1, 3.3}
What distinguishes a natural number from an integer?
What distinguishes a natural number from an integer?
- Natural numbers include fractions, while integers do not.
- Natural numbers can be negative, while integers cannot.
- Natural numbers must be positive, while integers can be negative or zero. (correct)
- Natural numbers include zero, while integers do not.
Which of the following statements is correct regarding prime numbers?
Which of the following statements is correct regarding prime numbers?
How does a square number differ from a cube number?
How does a square number differ from a cube number?
According to the order of operations (BIDMAS/BODMAS), which operation should be performed first in the following expression: $5 + 3 \times (6 - 2)^2 \div 4$?
According to the order of operations (BIDMAS/BODMAS), which operation should be performed first in the following expression: $5 + 3 \times (6 - 2)^2 \div 4$?
What is the purpose of the order of operations (BIDMAS/BODMAS) in mathematical expressions?
What is the purpose of the order of operations (BIDMAS/BODMAS) in mathematical expressions?
Simplify the expression: $(10 + 5) \div 3 - 2 \times 2$.
Simplify the expression: $(10 + 5) \div 3 - 2 \times 2$.
Which step comes immediately before addition in the order of operations (BIDMAS/BODMAS)?
Which step comes immediately before addition in the order of operations (BIDMAS/BODMAS)?
Why is it important to perform operations of the same rank (e.g., multiplication and division) from left to right?
Why is it important to perform operations of the same rank (e.g., multiplication and division) from left to right?
What is the Highest Common Factor (HCF) of two or more numbers?
What is the Highest Common Factor (HCF) of two or more numbers?
What is the significance of finding the Lowest Common Multiple (LCM) of two numbers in practical applications?
What is the significance of finding the Lowest Common Multiple (LCM) of two numbers in practical applications?
What is the HCF of 24 and 36?
What is the HCF of 24 and 36?
Determine the LCM of 15 and 20.
Determine the LCM of 15 and 20.
Why do we use prime factorization to find both HCF and LCM?
Why do we use prime factorization to find both HCF and LCM?
If $a^m = a^n$, what can we conclude?
If $a^m = a^n$, what can we conclude?
Simplify $(3^2)^3 \div 3^4$.
Simplify $(3^2)^3 \div 3^4$.
Which of the following is equivalent to $\sqrt{a} \times \sqrt{b}$?
Which of the following is equivalent to $\sqrt{a} \times \sqrt{b}$?
Express 0.000075 in standard form.
Express 0.000075 in standard form.
What does $a^0$ equal, assuming $a \neq 0$?
What does $a^0$ equal, assuming $a \neq 0$?
What is the cube root of 64?
What is the cube root of 64?
Express 2345 in standard form.
Express 2345 in standard form.
What is the square root of 81?
What is the square root of 81?
Simplify $\frac{\sqrt{16}}{\sqrt{4}}$
Simplify $\frac{\sqrt{16}}{\sqrt{4}}$
What is another way to express $5^{-2}$?
What is another way to express $5^{-2}$?
Evaluate the expression: $5 \times 2^3 - 3 \times (8 - 4) \div 2$.
Evaluate the expression: $5 \times 2^3 - 3 \times (8 - 4) \div 2$.
In standard form $a \times 10^n$, what condition must 'a' satisfy?
In standard form $a \times 10^n$, what condition must 'a' satisfy?
Find the HCF of 48 and 72.
Find the HCF of 48 and 72.
What is the simplified form of $\frac{a^5 \times a^{-2}}{a^2}$?
What is the simplified form of $\frac{a^5 \times a^{-2}}{a^2}$?
Flashcards
Natural Number
Natural Number
Whole numbers greater than 0. Denoted by 'N'.
Prime Numbers
Prime Numbers
Numbers with only two factors: 1 and the number itself.
Square Number
Square Number
The result when a number is multiplied by itself.
Cube Number
Cube Number
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Integers
Integers
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Rational Number
Rational Number
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Irrational Number
Irrational Number
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Real Number
Real Number
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Order of Operations (BIDMAS/BODMAS)
Order of Operations (BIDMAS/BODMAS)
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Highest Common Factor (HCF)
Highest Common Factor (HCF)
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Lowest Common Multiple (LCM)
Lowest Common Multiple (LCM)
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Prime Factorization
Prime Factorization
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Power (Exponent)
Power (Exponent)
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Root
Root
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Standard Form
Standard Form
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Product of Powers Rule
Product of Powers Rule
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Quotient of Powers Rule
Quotient of Powers Rule
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Power of a Power Rule
Power of a Power Rule
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Power of a Product Rule
Power of a Product Rule
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Zero Exponent Rule
Zero Exponent Rule
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Negative Exponent Rule
Negative Exponent Rule
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Product of Roots Property
Product of Roots Property
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Quotient of Roots Property
Quotient of Roots Property
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Fractional Exponent as Root
Fractional Exponent as Root
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Study Notes
Types of Numbers
- Natural numbers are whole numbers greater than 0, such as 1, 2, 3, 4, 5…, denoted by "N".
- Prime numbers have only two factors: 1 and themselves, for example: 2, 3, 5, 7.
- 1 is not a prime number because it has only one factor.
- Square numbers result from multiplying a number by itself, e.g., 2 x 2 = 4.
- Cube numbers result from multiplying a number by itself twice, e.g., 2 x 2 x 2 = 8.
- Integers include all whole numbers, both negative and positive, including 0, for example: -4, -3, -2, -1, 0, 1, 2, 3, 4, denoted by "Z".
- Rational numbers can be expressed as a fraction, with a non-zero denominator.
- Rational numbers are either terminating decimals or recurring decimals, denoted by “Q”.
- 0.75 can be written as 6/8
- 0.1(recurring) can be written as 1/9.
- Irrational numbers cannot be represented as a/b, examples include the square root of 2, and π.
- Real numbers include all numbers: integers, natural, rational, and irrational numbers, denoted by "R".
Mathematical Operations
- The order of operations, remembered by the acronym BIDMAS or BODMAS, is used to solve mathematical expressions correctly.
- BIDMAS: Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction.
- Always work from left to right for operations of the same rank.
- Simplify inside brackets first.
Example
- To solve (6 + 2) ÷ 2², first, 6 + 2 = 8.
- Then calculate the Indices, 2² = 4.
- Next divide, 8 ÷ 4 = 2.
HCF and LCM
Highest Common Factor (HCF)
- The largest number that divides two or more integers without a remainder.
Steps to Find HCF
- Prime Factorisation Method
- Express each number as a product of prime factors.
- Identify common prime factors with the smallest power.
- Multiply these common prime factors to get the HCF.
- Example: 12 = 2² x 3 and 18 = 2 x 3².
- The common prime factors of 12 and 18 are 2 and 3.
- HCF = 2 x 3 = 6.
Lowest Common Multiple (LCM)
- The smallest number that is a multiple of two or more integers.
Steps to Find LCM
- Prime Factorisation Method
- Express each number as a product of prime factors.
- Use the highest power of each prime number from the factorizations.
- Multiply these factors to get the LCM.
- Example: 4 = 2² and 6 = 2 x 3.
- For 4 and 6, the highest powers of prime factors are 2² and 3.
- LCM = 2² x 3 = 4 x 3 = 12.
Example: Find the LCM and HCF of 63 and 168
- Prime Factorisation of 63: 63 = 3² x 7.
- Prime Factorisation of 168: 168 = 2³ x 3 x 7.
- Finding the HCF: Common prime factors are 3 and 7.
- HCF = 3 x 7 = 21.
- Finding the LCM: Highest powers of prime factors are 2³, 3², and 7.
- LCM = 2³ x 3² x 7 = 8 x 9 x 7 = 504.
Calculation of Powers and Roots
Powers
- A power indicates how many times a number is multiplied by itself.
- Example: 2¹ = 2, 2² = 4, 2³ = 8.
- In the expression, 2 is the base, and the numbers 1, 2, and 3 are the exponents.
7 Laws of Exponents
- aᵃ x aᵇ = aᵃ⁺ᵇ
- aᵃ ÷ aᵇ = aᵃ⁻ᵇ
- (aᵃ)ᵇ = aᵃˣᵇ
- (a/b)ᵇ = aᵇ/bᵇ
- a⁰ = 1
- (ab)ᵇ = aᵇbᵇ
- a⁻ᵇ = 1/aᵇ
Roots
Square root
- √a = b means a = b².
- √4 = 2 because 2² = 4.
Cube root
- ³√a = b means b³ = a.
- ³√8 = 2 because 2³ = 8.
Properties of Roots
- √a x √b = √(a x b)
- √a / √b = √(a/b)
- a^(1/b) = ᵇ√a
Standard Form
- Standard form is represented as a x 10ⁿ, where 1 ≤ |a| < 10
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