Everything covered in this lesson, in one place - useful for revision or printing.
\[ \text{Whole Numbers} \]
Welcome to Grade 8 Mathematics. In this lesson, we will master whole numbers. We will explore the properties of operations, find highest common factors and lowest common multiples, work with squares and cubes, and apply the correct order of operations.
\[ a+b=b+a \quad a \times b = b \times a \]
The commutative law means you can swap the order of numbers when adding or multiplying. Three plus five is the same as five plus three. Four times six equals six times four.
\[ (a+b)+c = a+(b+c) \]
The associative law means you can group numbers differently. The answer stays the same whether you add two and three first, or three and four first.
\[ a(b+c) = ab + ac \]
The distributive law is very powerful. Three times the bracket ten plus four equals three times ten plus three times four. This helps us do mental math quickly.
\[ a+0=a \quad a \times 1 = a \]
Finally, the identity properties. Adding zero changes nothing. Multiplying by one changes nothing. Zero is the additive identity, and one is the multiplicative identity.
\[ \text{Commutative, Associative, Distributive, Identity} \]
These four properties help us understand how numbers work together. They make calculations easier and more flexible.
\[ \text{HCF} = \text{Highest Common Factor} \]
The Highest Common Factor is the largest number that divides exactly into two or more numbers. The Lowest Common Multiple is the smallest number that is a multiple of two or more numbers.
\[ 12 = 2^2 \times 3 \quad 18 = 2 \times 3^2 \]
To find HCF and LCM, use prime factorization. Twelve is two squared times three. Eighteen is two times three squared.
\[ \text{HCF} = 2 \times 3 = 6 \]
For the HCF, we take the common prime factors with the lowest power: two times three equals six.
\[ \text{LCM} = 2^2 \times 3^2 = 36 \]
For the LCM, we take the highest power of every prime factor: two squared times three squared equals thirty-six.
\[ \text{HCF: lowest powers} \quad \text{LCM: highest powers} \]
Remember: HCF uses the lowest powers of common primes. LCM uses the highest powers of all primes.
\[ 5^2 = 5 \times 5 = 25 \]
Squaring a number means multiplying it by itself. Five squared is twenty-five.
\[ \sqrt{25} = 5 \]
The square root is the reverse operation. The square root of twenty-five is five.
\[ 3^3 = 3 \times 3 \times 3 = 27 \]
Cubing a number means multiplying it by itself three times. Three cubed is twenty-seven.
\[ \sqrt[3]{27} = 3 \]
The cube root of twenty-seven is three.
\[ \text{Squares and cubes are inverse to roots} \]
Squares and cubes are powers. Square roots and cube roots are the inverse operations that undo them.
\[ \textbf{B} - \text{Brackets} \]
When an expression has multiple operations, we must follow BODMAS. Brackets first.
\[ \textbf{O} - \text{Orders (Powers and Roots)} \]
Then Orders like powers and roots.
\[ \textbf{D/M} - \text{Division and Multiplication} \]
Then Division and Multiplication from left to right.
\[ \textbf{A/S} - \text{Addition and Subtraction} \]
Finally, Addition and Subtraction from left to right.
\[ 2 + 3 \times 4^2 = 50 \]
Let us solve two plus three times four squared. First, calculate the order: four squared is sixteen. Next, do the multiplication: three times sixteen is forty-eight. Finally, add two. The answer is fifty.
\[ \text{Always follow BODMAS} \]
Always follow the correct order: Brackets, Orders, Division/Multiplication, Addition/Subtraction.
\[ \text{Properties, HCF/LCM, Powers, BODMAS} \]
Excellent work. You have mastered the foundations of whole numbers. You understand the properties of operations, can find HCF and LCM using prime factors, work confidently with squares and cubes, and always apply BODMAS correctly. You are now ready to test your knowledge in the quiz.