Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Simplify expressions with exponents and surds.
• Solve quadratic equations and inequalities.
• Solve simultaneous equations (one linear, one quadratic).
• Solve exponential equations.
• Apply these concepts to word problems.

Lesson summary

This week marks the crucial beginning of our focused revision for the final Grade 12 Mathematics examination. We will concentrate on foundational topics crucial for success in more advanced concepts. A strong grasp of these basics is essential not just for passing the exam, but for future studies and careers. Mathematics, even at this level, provides tools for critical thinking, problem-solving, and logical reasoning applicable to a wide range of situations, from managing personal finances to understanding complex technological advancements and national budget allocations.

Lesson notes

2.1 Exponents Definition: An exponent indicates how many times a number (the base) is multiplied by itself. `a^n` means 'a' multiplied by itself 'n' times.

Laws of Exponents: These are fundamental rules for simplifying expressions involving exponents. Remember why they work, not just what they are.

Product of Powers: `a^m a^n = a^(m+n)` (When multiplying powers with the same base, add the exponents). Why? `a^m` is 'a' multiplied by itself 'm' times, and `a^n` is 'a' multiplied by itself 'n' times. Multiplying them together means 'a' is multiplied by itself a total of 'm+n' times.

Quotient of Powers: `a^m / a^n = a^(m-n)` (When dividing powers with the same base, subtract the exponents). Why?* Similar to above, division cancels out common factors of 'a'.

Power of a Power: `(a^m)^n = a^(mn)` (When raising a power to another power, multiply the exponents). Why? `(a^m)^n` is `a^m` multiplied by itself 'n' times, and each `a^m` is 'a' multiplied by itself 'm' times. So, 'a' is multiplied by itself a total of `m*n` times.

Power of a Product: `(ab)^n = a^n b^n` (The power of a product is the product of the powers).

Power of a Quotient: `(a/b)^n = a^n / b^n` (The power of a quotient is the quotient of the powers).

Zero Exponent: `a^0 = 1` (Any non-zero number raised to the power of 0 is 1). Why?* Consider `a^m / a^m = 1`. Using the quotient rule, `a^m / a^m = a^(m-m) = a^0`.

Therefore, `a^0 = 1`.

Negative Exponent: `a^(-n) = 1 / a^n` (A negative exponent indicates the reciprocal of the base raised to the positive exponent). Why?* Consider `a^0 / a^n = 1 / a^n`. Using the quotient rule, `a^0 / a^n = a^(0-n) = a^(-n)`.

Therefore, `a^(-n) = 1 / a^n`.

Example 1: Simplify: `(2x^3y^(-2))^2 * (3x^(-1)y^4)` Solution: Distribute the outer exponent: `(4x^6y^(-4)) * (3x^(-1)y^4)` Multiply the coefficients: `12x^6y^(-4)x^(-1)y^4` Combine like terms by adding exponents: `12x^(6-1)y^(-4+4)` Simplify: `12x^5y^0` `y^0 = 1`, so the final answer is `12x^5` 2.2 Surds Definition: A surd is an irrational root of a rational number. It cannot be expressed as a simple fraction.

Examples: √2, √3, √5. √4 is NOT a surd because √4 =

2. Properties of Surds: `√(ab) = √a * √b` `√(a/b) = √a / √b` `(√a)^2 = a` `√a √a = a` Rationalizing the Denominator: To remove a surd from the denominator of a fraction, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of `a + √b` is `a - √b`, and vice-versa.

Example 2: Simplify: `√27 + 2√12 - √75` Solution: Express each surd in its simplest form: `√27 = √(93) = √9 * √3 = 3√3` `√12 = √(43) = √4 * √3 = 2√3` `√75 = √(253) = √25 * √3 = 5√3` Substitute these simplified forms back into the original expression: `3√3 + 2(2√3) - 5√3` Simplify: `3√3 + 4√3 - 5√3` Combine like terms: `(3 + 4 - 5)√3 = 2√3` Example 3: Rationalize the denominator: `3 / (2 + √5)` Solution: Identify the conjugate of the denominator: The conjugate of `2 + √5` is `2 - √5`. Multiply both numerator and denominator by the conjugate: `(3 (2 - √5)) / ((2 + √5) (2 - √5))` Expand: `(6 - 3√5) / (4 - 2√5 + 2√5 - 5)` Simplify: `(6 - 3√5) / (-1)` Multiply numerator by -1: `-6 + 3√5` or `3√5 - 6` 2.3 Equations and Inequalities Quadratic Equations: An equation of the form `ax^2 + bx + c = 0`, where a ≠

0. Methods of Solving: Factorization: Express the quadratic expression as a product of two linear factors. Set each factor equal to zero and solve for x.

Completing the Square: Rewrite the quadratic equation in the form `(x + p)^2 = q`.

Quadratic Formula: `x = (-b ± √(b^2 - 4ac)) / (2a)` Quadratic Inequalities: An inequality of the form `ax^2 + bx + c > 0`, `ax^2 + bx + c 0` Solution: Solve the corresponding quadratic equation: `x^2 - 3x - 4 = 0` Factorize: `(x - 4)(x + 1) = 0` Critical values: `x = 4` or `x = -1` Draw a number line and mark the critical values: ``` ``` Choose test values: `x 0` (True) `-1 0` (False) `x > 4`: Let `x = 5`. `(5)^2 - 3(5) - 4 = 25 - 15 - 4 = 6 > 0` (True)

Solution set: `x 4`.

In interval notation: `(-∞, -1) ∪ (4, ∞)` Example 6: Solve the simultaneous equations: `y = x + 1` `x^2 + y^2 = 5` Solution: Substitute `y = x + 1` into the second equation: `x^2 + (x + 1)^2 = 5` Expand and simplify: `x^2 + x^2 + 2x + 1 = 5` `2x^2 + 2x - 4 = 0` Divide by 2: `x^2 + x - 2 = 0` Factorize: `(x + 2)(x - 1) = 0` Solve for x: `x = -2` or `x = 1` Substitute these values back into `y = x + 1`: If `x = -2`, then `y = -2 + 1 = -1` If `x = 1`, then `y = 1 + 1 = 2` Solutions: `(-2, -1)` and `(1, 2)` Example 7: Solve the exponential equation: `2^(x+1) = 8` Solution: Recognize that 8 can be written as 2^3.