GRADE 6 ADVANCE MATHEMATICS - NOTATIONST
- 1. MATHEMATICES
- 2. 1. Notation of squares and Cubes • A concise mathematical way of expressing repeated multiplication of a number by itself • Understanding the notation of squares and cubes is essential in mathematics as they often appear in algebra, geometry, and other scientific disciplines.
- 3. a. Square Notation • The square of a number means multiplying the number by itself once. It is denoted by the exponent 2. • If is any number, then the square of is written as ^2. 𝑥 𝑥 𝑥 • In geometry, ^2 represents the area of a square with side 𝑥 length . 𝑥 EXAMPLE 𝟓𝟐 =𝟓× 𝟓=𝟐𝟓 𝟑𝟐 =𝟑 × 𝟑=𝟗
- 4. b. Cube Notation • The cube of a number means multiplying the number by itself two more times. It is denoted by the exponent 3. • If is any number, then the cube of is written as . 𝑥 𝑥 • In geometry, represents the volume of a cube with side length . 𝑥 EXAMPLE 𝟐𝟑 =𝟐×𝟐×=𝟖 𝟒𝟑 =𝟒×𝟒×𝟒=𝟔𝟒
- 5. Properties of Squares and Cubes SQUARES • Always non-negative: • Even exponents result in squares.
- 6. Properties of Squares and Cubes CUBES • Preserve the sign of the base: A negative number cubed remains negative, while a positive number cubed remains positive. • Odd exponents result in cubes
- 7. 2. Decanomial Square • A square grid where the side lengths correspond to the numbers from 1 to 10, and each cell in the grid represents the product of the corresponding row and column numbers. For example, the cell in the 3rd row and 4th column represents 3×4. • Helps in understanding multiplication and squaring.
- 8. Example of Decanomial Square
- 9. 3. Transformation of Squares • Involves modifying or reconfiguring square expressions to simplify, expand, or manipulate them for mathematical or geometric purposes. • Transformations can include algebraic operations (e.g., completing the square, factorization) or geometric reinterpretations (e.g., rearranging areas).
- 10. a. Expanding Squares • When a binomial or polynomial is squared, it can be expanded using the distributive property or formulas such as: EXPAND EXAMPLE +6x+9
- 11. b. Factoring Squares • Factoring reverses the expansion process. A perfect square trinomial can be written as a square of a binomial. FACTOR +10x+25 EXAMPLE +10x+25 =
- 12. c. Difference of Squares • The difference of squares formula is used to simplify expressions involving the subtraction of two square terms: SIMPLIFY - 9 EXAMPLE -9=(x+3)(x-3)
- 13. 4. Sum of Squares and Cubes • The sum of squares and sum of cubes are mathematical concepts that involve the summation of squared or cubed terms.
- 14. a. Sum of Squares • The sum of squares involves summing the squares of terms. It is often used to analyze patterns, solve equations, and perform numerical analysis.
- 15. Find the sum of squares of the first 4 even numbers. EXAMPLE
- 16. Calculate the sum of squares of the first 5 natural numbers. EXAMPLE
- 17. b. Sum of Cubes • The sum of cubes involves summing the cubes of terms. It is closely related to the formula for the square of sums.
- 18. Calculate the sum of cubes of the first 4 natural numbers. EXAMPLE
- 19. 5. Difference of Squares and Cubes • The difference of squares and difference of cubes are mathematical identities that simplify expressions where two terms are subtracted, and each term is either squared or cubed. • The difference of squares formula states: This identity allows the factorization of expressions involving the subtraction of two squared terms.
- 20. a. Difference of Squares • The difference of squares formula states: • This identity allows the factorization of expressions involving the subtraction of two squared terms.
- 21. Simplify EXAMPLE
- 22. Solve EXAMPLE
- 23. Solve EXAMPLE
- 24. b. Difference of Cubes • The difference of cubes formula is: • This formula helps to factorize expressions involving the subtraction of two cubed terms.
- 25. Simplify EXAMPLE
- 26. Solve EXAMPLE
- 27. Simplify EXAMPLE
- 28. 6. Laws of Exponents
- 29. a. Power Rule • The power rule of exponents is a fundamental property in algebra that deals with the situation where a base raised to an exponent is itself raised to another exponent. • This means that when raising a power to another power, you multiply the exponents. Special Case:
- 32. b. Product Rule • The product rule of exponents is a basic property in algebra that simplifies the multiplication of two expressions with the same base. This rule makes calculations involving exponents straightforward and efficient. • This means that when multiplying two powers with the same base, you add the exponents. • Special Case:
- 35. c. Quotient Rule • The quotient rule of exponents helps simplify expressions where powers with the same base are divided. It allows us to efficiently handle division by subtracting the exponents. • This means that when dividing two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
- 39. Simplify EXAMPLE
- 40. Zero Rule • The zero exponent rule states that any nonzero number raised to the power of zero is always equal to 1. • This rule is based on the pattern and logic of dividing powers with the same base. When exponents are subtracted, if the numerator and denominator have the same power, the result simplifies to .
- 42. Simplify EXAMPLE
- 44. 7. Cubing a Binomial • Cubing a binomial refers to raising a binomial expression (the sum or difference of two terms) to the power of three. • In mathematical terms, a binomial is an expression of the form ( + ) 𝑎 𝑏 (a+b) or ( )(a b), and cubing means multiplying it by itself three 𝑎−𝑏 − times.
- 45. Formula of Cubing a Binomial
- 46. Explanation of the Formula
- 47. Expand EXAMPLE
- 48. Expand EXAMPLE
- 49. Expand EXAMPLE
- 50. Expand EXAMPLE
- 51. 8. Cubing a Trinomial • Cubing a trinomial involves raising a trinomial expression (an algebraic expression with three terms) to the power of three. In other words, you multiply the trinomial by itself three times.
- 52. Formula of Cubing a Trinomial
- 53. Steps to Cube a Trinomial
- 54. Expand EXAMPLE
- 55. Expand EXAMPLE
- 56. Expand EXAMPLE