Math Prep 1 Second Term Review Flashcards

Unit 1: Powers, Exponents and Roots

• Repeated Multiplication and Exponential Form: The operation is expressed as $a^n$, where $a$ is the base and $n$ is the exponent (indicating the number of times the factor is repeated).

• Even and Odd Exponents of Negative Base:     * If $m$ is even: $(-a)^m = a^m$ (e.g., $(-3)^2 = 9$).     * If $m$ is odd: $(-a)^m = -a^m$ (e.g., $(-3)^3 = -27$).

• Laws of Exponents ($a, b \neq 0$):     * Multiplication: $a^m \times a^n = a^{m+n}$     * Division: $a^m \div a^n = a^{m-n}$     * Power of Zero: $a^0 = 1$     * Negative Exponents: $a^{-n} = \frac{1}{a^n}$ (the multiplicative inverse of $a^n$).

• Scientific Notation: Expresses numbers in the form $a \times 10^n$, where 1 \le |a| < 10 and $n \in \mathbb{Z}$.

• Perfect Squares and Square Roots: The positive square root of a perfect square $a$ is denoted by $\sqrt{a}$. If $x^2 = a$, then $x = \pm \sqrt{a}$.

• Perfect Cubes and Cube Roots: The cube root of a number $a$ is denoted by $\sqrt[3]{a}$. Unlike square roots, the cube root preserves the original sign of the number ($\sqrt[3]{-8} = -2$).

Unit 2: Algebra and Inequalities

• Inequalities: Mathematical expressions separated by <, >, \le, \text{ or } \ge .     * Properties: Adding or subtracting the same number does not change the inequality direction. Multiplying or dividing by a negative number reverses the direction of the inequality sign.

• Multiplying Algebraic Terms: Multiply coefficients and add exponents of like variables ($ax^m \times bx^n = (ab)x^{m+n}$).

• Product of Algebraic Expressions:     * Square of a Binomial: $(a \pm b)^2 = a^2 \pm 2ab + b^2$     * Sum and Difference of Two Terms: $(a + b)(a - b) = a^2 - b^2$

• Division: Divide coefficients and subtract exponents of like variables ($\frac{ax^m}{bx^n} = (\frac{a}{b})x^{m-n}$).

Unit 3: Geometry and Measurement

• Area Formulas:     * Rhombus: $A = \frac{1}{2} \times d_1 \times d_2$ (where $d_1, d_2$ are diagonals).     * Square (by diagonal): $A = \frac{1}{2} d^2$.     * Trapezium:         * $A = \frac{1}{2} (b_1 + b_2) \times h$         * The length of the middle base: $L = \frac{1}{2} (b_1 + b_2)$         * Area in terms of middle base: $A = L \times h$

• Geometrical Constructions: Essential skills include bisecting angles, bisecting line segments, and drawing triangles using SSS, SAS, or ASA criteria.

• Geometrical Transformations: Transformations that preserve congruence (isometries):     * Reflection: In $X$-axis: $(x, y) \rightarrow (x, -y)$; in $Y$-axis: $(x, y) \rightarrow (-x, y)$.     * Translation: Moves a shape by $(a, b)$: $(x, y) \rightarrow (x + a, y + b)$.     * Rotation (around origin):         * $R(O, 90^\circ)$: $(x, y) \rightarrow (-y, x)$         * $R(O, -90^\circ)$: $(x, y) \rightarrow (y, -x)$         * $R(O, \pm 180^\circ)$: $(x, y) \rightarrow (-x, -y)$

Unit 4: Probabilities

• Random Experiment: An experiment where all possible outcomes are known in advance, but the specific outcome is unpredictable before performance.

• Sample Space ($S$): The set of all possible outcomes. The frequency is $n(S)$.

• Event Types:     * Certain Event: Includes all elements in $S$; probability $P(S) = 1$.     * Impossible Event: Set is empty ($\phi$); probability $P(\phi) = 0$.     * Simple Event: Subset containing one outcome.

• Theoretical Probability: calculated as $P(A) = \frac{\text{number of outcomes in event A}}{\text{total number of outcomes in S}}$.

• Experimental Probability: calculated as $\frac{\text{number of times the event occurs}}{\text{total number of trials}}$.

Repeated Multiplication and Exponential Form: The operation is expressed as $a^n$, where $a$ is the base and $n$ is the exponent (indicating how many times the factor is repeated).

Exercise: Simplify: $4^3$.

Even and Odd Exponents of Negative Base: If $m$ is even: $(-a)^m = a^m$ (e.g., $(-3)^2 = 9$). If $m$ is odd: $(-a)^m = -a^m$ (e.g., $(-3)^3 = -27$).

Exercise: Evaluate: $(-7)^4$ and $(-7)^5$.

Laws of Exponents ($a, b \neq 0$):

• Multiplication: $a^m \times a^n = a^{m+n}$

• Division: $a^m \div a^n = a^{m-n}$

• Power of Zero: $a^0 = 1$

• Negative Exponents: $a^{-n} = \frac{1}{a^n}$ (the multiplicative inverse of $a^n$).

Exercise: Simplify: $5^2 \times 5^3$ and $\frac{6^5}{6^2}$.

Scientific Notation: Expresses numbers in the form $a \times 10^n$, where $1 \le |a|$.

Exercise: Convert the number 3000 into scientific notation.