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Questions and Answers
Which expression simplifies to $x^6$ when using index rules?
Which expression simplifies to $x^6$ when using index rules?
What is the highest common factor (HCF) of the numbers 36 and 60?
What is the highest common factor (HCF) of the numbers 36 and 60?
Which of the following numbers can be expressed as a product of prime numbers?
Which of the following numbers can be expressed as a product of prime numbers?
If you have the equation $3x + 4 = 19$, what is the value of x?
If you have the equation $3x + 4 = 19$, what is the value of x?
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Using BIDMAS, what is the value of $2 + 3 \times (8 \div 4)$?
Using BIDMAS, what is the value of $2 + 3 \times (8 \div 4)$?
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Which of the following expressions is equivalent to $x^5 \cdot x^3$?
Which of the following expressions is equivalent to $x^5 \cdot x^3$?
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What is the LCM of 12, 15, and 20?
What is the LCM of 12, 15, and 20?
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If $y = 2x + 3$, and $x = 5$, what is the value of $y$?
If $y = 2x + 3$, and $x = 5$, what is the value of $y$?
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Which expression represents the factorization of $x^2 - 4$?
Which expression represents the factorization of $x^2 - 4$?
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Which of the following equations represents the statement: 'Three times a number decreased by 5 equals 7'?
Which of the following equations represents the statement: 'Three times a number decreased by 5 equals 7'?
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Study Notes
Prime Factorisation
- Every whole number greater than 1 can be written as a product of prime numbers
- Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves
- To write a whole number as a product of primes, divide the number by the smallest prime number that divides it, and repeat the process until you reach a prime number
- For example, 24 can be written as 2 x 2 x 2 x 3, or 2³ x 3
Highest Common Factor (HCF)
- The HCF of two or more numbers is the highest number that divides into all of them without leaving a remainder
- To find the HCF, write each number as a product of prime numbers, then multiply all the common prime factors together
- For example, the HCF of 12 and 18 is 6, because 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3
Lowest Common Multiple (LCM)
- The LCM of two or more numbers is the lowest number that is a multiple of all of them
- To find the LCM, write each number as a product of prime numbers, then multiply together the highest powers of all prime factors
- For example, the LCM of 12 and 18 is 36, because 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3
Order of Operations (BIDMAS)
- BIDMAS stands for Brackets, Indices, Division and Multiplication, Addition and Subtraction
- This is the order in which operations should be performed in a mathematical expression
- For example, in the expression 2 + 3 x 4, the multiplication should be performed before the addition, so the answer is 14, not 20
Indices
- An index is a number that indicates how many times a base number is multiplied by itself
- For example, 2⁴ means 2 x 2 x 2 x 2 = 16
- Index rules include:
- xᵐ x xⁿ = xᵐ⁺ⁿ
- xᵐ ÷ xⁿ = xᵐ⁻ⁿ
- (xᵐ)ⁿ = xᵐⁿ
Expressions as Powers
- Any expression can be written as a power of one number using index rules
- For example, x²y³ can be written as (xy)⁵
Forming Equations
- To form an equation from a word problem, identify the unknown quantity and represent it with a variable
- Then, write down the relationships between the quantities in the problem using mathematical symbols
- For example, if the problem says "The sum of two numbers is 10", you can write this as x + y = 10, where x and y represent the two numbers
Solving Linear Equations
- A linear equation is an equation that can be written in the form ax + b = 0, where a and b are constants and x is a variable
- To solve a linear equation, isolate the variable x on one side of the equation using inverse operations
- For example, to solve the equation 2x + 3 = 7, we can subtract 3 from both sides to get 2x = 4, and then divide both sides by 2 to get x = 2
Simplifying Expressions
- An expression is a combination of numbers and variables connected by mathematical operations
- To simplify an expression, combine like terms and use index rules
- For example, the expression 3x + 2y - x + 5y can be simplified to 2x + 7y
Substitution
- Substitution is the process of replacing a variable in an expression or equation with its value
- For example, if x = 2, then the expression 3x + 1 can be simplified by substituting 2 for x, giving 3(2) + 1 = 7
Factorising
- Factorising is the process of writing an expression as a product of its factors
- To factorise an expression, look for common factors and use the distributive property
- For example, the expression 2x² + 4x can be factorised by taking out a common factor of 2x, giving 2x(x + 2)
Prime Factorization
- Expressing a number as a product of prime numbers.
- A prime number is a number only divisible by 1 and itself.
Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
- HCF is the highest number that divides two or more numbers.
- LCM is the smallest number that is a multiple of two or more numbers.
BIDMAS
- An order of operations for calculations, standing for:
- Brackets
- Indices
- Division and Multiplication (from Left to Right)
- Addition and Subtraction (from Left to Right)
Indices
- Indices show how many times a number is multiplied by itself.
- Simplify expressions using index rules:
- am x an = am+n
- am / an = am-n
- (am)n = am x n
Expressions as a Power of One Number
- Rewrite expressions with repeated multiplications using index notation.
Forming Equations
- Translate written problems into mathematical equations, using variables to represent unknown quantities.
Solving Linear Equations
- Finding the value of the unknown variable in an equation.
- Use inverse operations to isolate the variable.
Simplifying Expressions
- Combine like terms by adding or subtracting their coefficients.
- Apply the rules of indices and other algebraic operations.
Substitution
- Replace variables in an expression or equation with their given values.
Factorising
- Expressing an expression as a product of its factors.
- Finding common factors to simplify expressions.
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Description
Explore the concepts of prime factorisation, highest common factor (HCF), and lowest common multiple (LCM) in this quiz. Learn how to break down numbers into their prime factors, and find HCF and LCM using prime factorisation methods. Test your understanding through practice questions and examples.
