Summary of Grade 8 Mathematics Control Test

## Introduction This study material reviews a range of skills from a school-level mathematics test: exponents, simplifying expressions, number patterns, functions, and basic algebraic manipulation. Work through the explanations, examples and practice questions to build confidence in solving test-style problems. > Definition: An exponent shows how many times a base is multiplied by itself, for example $a^n$ means multiply $a$ by itself $n$ times. ## Exponents and Laws ### Basic ideas - If the base is $a$ and the exponent is $n$, then $a^n$ means $a\times a\times\dots\times a$ ($n$ factors). - Special cases: $a^0 = 1$ when $a\ne 0$, and $a^1 = a$. > Definition: A negative exponent means reciprocal: $a^{-n} = \dfrac{1}{a^n}$ for $a\ne 0$. ### Laws of exponents (small digestible rules) - Product rule: $a^m\,a^n = a^{m+n}$ - Quotient rule: $\dfrac{a^m}{a^n} = a^{m-n}$ for $a\ne 0$ - Power of a power: $(a^m)^n = a^{mn}$ - Power of a product: $(ab)^n = a^n b^n$ Table: Rule and quick example | Rule | Example | | --- | --- | | Product | $2^3\,2^2 = 2^{3+2} = 2^5$ | | Quotient | $\dfrac{5^4}{5^2} = 5^{4-2} = 5^2$ | | Power of a power | $(3^2)^3 = 3^{2\cdot 3} = 3^6$ | | Power of a product | $(2\cdot 5)^2 = 2^2\cdot 5^2$ | ### Examples 1. Simplify $(-1)^{27}\cdot(-1)^{-2}$. The exponents add: $(-1)^{27-2} = (-1)^{25} = -1$ because odd exponent of $-1$ is $-1$. 2. Simplify $\left(2^2\right)\left(2^2\right)\left(2^2\right)$. Multiply powers with same base: $2^{2+2+2}=2^6$. 3. Simplify $\dfrac{m^4}{m^6}$. Use quotient rule: $m^{4-6} = m^{-2} = \dfrac{1}{m^2}$ for $m\ne 0$. ### Practice question (with steps) Simplify: $(-127)^{-2}\times(-5)^2$. - Note that $(-127)^{-2} = \dfrac{1}{(-127)^2}$ and $(-5)^2 = 25$. - So the product is $\dfrac{25}{(-127)^2}$. - If numeric form is required, compute $(-127)^2 = 127^2 = 16129$, so result $\dfrac{25}{16129}$. Fun fact: Negative exponents are just a compact way to write reciprocals, so $a^{-3} = \dfrac{1}{a^3}$ for any nonzero $a$. ## Simplifying Algebraic Expressions ### Like and unlike terms - Like terms have the same variable part (same letters and same exponents), e.g. $2a$ and $-5a$ are like terms; $2a$ and $-5a^2$ are unlike. > Definition: "Like terms" are terms that have identical variable factors; only coefficients differ. ### Combining like terms - Example: $2a - 4a^2 - 8a$ contains like terms $2a$ and $-8a$; combine them: $(2a-8a)-4a^2 = -6a - 4a^2$. ### Expanding brackets - Use distribution: $-6(3a-2b) + 6a - b$. Step 1: Expand $-6(3a-2b) = -18a +12b$. Step 2: Add remaining terms: $-18a +12b +6a - b = (-18a+6a) + (12b-b) = -12a +11b$. ### Practice value substitution - Evaluate $2a - 4a^2 - 8a$ for $a=-1$. Combine first: $2a-8a = -6a$, so expression $-6a - 4a^2$. Now substitute: $-6(-1)-4(-1)^2 = 6 -4 = 2$. Did you know that substituting values into algebraic expressions is the same process used in computer programming when evaluating formulas for given inputs? ## Numeric and Geometric Patterns ### Understanding pattern rules - Look for how the figure or number changes between steps. Often there is a linear rule: number of elements = $an + b$ where $n$ is the pattern number. > Definition: A term-to-term rule expresses how to get from one term to the next; an algebraic rule expresses the $n$th term directly in terms of $n$. ### Example with squares pattern Suppose the first three patterns have these numbers of squares: 1st = 3, 2nd = 6, 3rd = 9 (example sequence that increases by 3). - An algebraic rule is $S(n) = 3n$. - For $n=6$: $S(6)=3\cdot 6=18$ squares. - Which $n$ gives $S(n)=1025$? Solve $3n=1025$ so $n=\dfrac{1025}{3}$ (not an integer, so no whole pattern number produces 1025 squares). ### Steps to find a rule from diagrams 1. Count squares for first few patterns. 2. Find constant difference (arithmetic) or ratio (geometric). 3. Fit linear rule $an+b$ if differences are constant. ## Functions and Tables ### Reading a function table - A funct