Algebra

Multiplying and Dividing Exponents - Rules, Examples

Multiplying exponents with the same base adds them: $x^m \cdot x^n = x^{m + n}$. Dividing subtracts them: $\tfrac{x^m}{x^n} = x^{m - n}$. The two rules require the same base on both sides — without that, no exponent rule applies. This article covers the product rule, the quotient rule, negative and fractional exponents, the three errors that cost marks, and worked examples.

Explicit Formulas for Sequences - Methods, Examples

An explicit formula for a sequence gives the $n$-th term $a_n$ directly as a function of $n$ — no need to compute the previous terms first. For an arithmetic sequence, the formula is $a_n = a_1 + (n-1)d$. For geometric, $a_n = a_1 \cdot r^{n-1}$. This article covers the derivation, the four most-tested sequence types, worked examples, the three off-by-one mistakes that cost marks, and the mathematicians who built the formulas.

Solve Matrices - Gaussian Elimination, Inverse, Cramer's Rule

Squares and Square Roots - Rules, Properties, Examples

Squares multiply a number by itself; square roots reverse the operation. Learn the rules, perfect squares, irrationality, simplification, and the most common mistakes.

Zeros of a Function - Definition, Methods, Examples

A zero of a function $f$ is any input value $x$ for which $f(x) = 0$. On a graph, the zeros are the $x$-coordinates where the curve crosses or touches the $x$-axis. This article covers the formal definition, the four standard methods for finding zeros multiplicity of zeros, the three common errors, and the historical story that connects zeros to the rise of polynomial equations.

Adding Exponents - Rules, Methods, Examples

Adding exponents follows two rules — only like terms combine, and exponents themselves don't add. Worked examples for same base, different bases, and fractional powers.

Standard to Vertex Form - Conversion, Formula, Examples

Converting from standard form to vertex form turns $ax^2 + bx + c$ into $a(x - h)^2 + k$, where $(h, k)$ is the parabola's vertex. The two reliable methods are completing the square (always works, builds intuition) and the shortcut formula $h = -\tfrac{b}{2a}$, $k = c - \tfrac{b^2}{4a}$. This article covers both methods, three worked examples, the three mistakes that cost marks, the real-world WHY behind the vertex.

Relations in Math - Definition, Types, Examples

A relation in math is a rule that links each element of one set to one or more elements of another set, written as a collection of ordered pairs. This article covers the formal definition, domain and range, the eight standard types (empty, universal, identity, reflexive, symmetric, transitive, equivalence, inverse), how a relation differs from a function, and the three mistakes that cost students marks every term.

Types of Polynomials — Classification with Examples

Polynomials are classified in two independent ways: by degree (constant, linear, quadratic, cubic, quartic, quintic, and higher) and by number of terms (monomial, binomial, trinomial, and beyond). This article walks through every type, gives three worked examples spanning Quick to Stretch, lists the mistakes that cost marks, and credits the people who built the classification.

Factoring Cubic Polynomials — Methods & Examples

To factor a cubic polynomial, three reliable methods cover almost every case — the Rational Root Theorem, the grouping method, and the sum/difference of cubes identities. This article walks through all three with three worked examples spanning Quick to Stretch, the mistakes that cost marks, and the people who built the framework.

Properties of Equality — List, Definitions, Examples

The properties of equality are the nine algebraic rules that say if you do the same thing to both sides of an equation, the equation stays true. This article lists all nine, gives three worked examples from Quick to Stretch, walks through the mistakes that cost marks, and credits the mathematicians who codified the axioms.

Standard Form of a Polynomial — Rules and Examples

The standard form of a polynomial writes terms in descending order of degree, with no like terms and no zero-coefficient terms. This article covers the two rules of standard form, three worked examples, the common slips, and a side-by-side comparison of polynomial types — monomial through quintic — organised by degree and number of terms.

Rational Exponents — Rules, Properties, Examples

A rational exponent is a fractional power: $a^{p/q}$ equals the $q$-th root of $a^p$. This article covers the conversion between rational exponents and radicals, three worked examples, the common slips, and a full cheat sheet of exponent rules — product, quotient, power-of-a-power, zero, negative, and rational — in one reference table.

Complement of a Set — Definition, Symbol, Examples

The complement of a set $A$, written $A'$ (or $A^c$, or $\overline{A}$), is the set of all elements in the universal set $U$ that are not in $A$. This article covers the definition, the symbol, three worked examples, the properties, and the common slips students make when working with complements.

Intersection of Sets — Symbol, Definition, Examples

The intersection of sets $A$ and $B$, written $A \cap B$, is the set of elements that are in both $A$ and $B$. This article covers the formal definition, the symbol, properties, three worked examples, the common slips, and a side-by-side comparison with union, set difference, and complement.

Cubic Polynomials — Definition, Properties, Examples

A cubic polynomial is a polynomial of degree 3, of the form $p(x) = ax^3 + bx^2 + cx + d$ with $a \neq 0$. This article covers the definition, the shape and roots of its graph, three worked examples, the most common slips, and the 16th-century rivalry between del Ferro, Tartaglia, and Cardano that produced the cubic formula.

Multiplication of Vectors — Dot & Cross Product Explained

Two non-zero vectors can be multiplied in two ways: the dot product (a scalar), which measures how much one vector projects onto the other; and the cross product (a vector), which produces a third vector perpendicular to both with magnitude equal to the parallelogram area they span.

Roots of Quadratic Equation — Types and Formulas

The roots of a quadratic equation $ax^2 + bx + c = 0$ are the values of $x$ that satisfy it. The discriminant $D = b^2 - 4ac$ classifies them into three types — real and distinct, real and equal, or complex. This article covers the quadratic formula, Vieta's sum-and-product relations, three worked examples, and a 1,400-year history from Brahmagupta to Cardano.

Reflexive Property of Equality — Definition and Examples

The reflexive property of equality says every quantity equals itself — $a = a$ for every $a$. This article covers the definition, three worked examples (algebra, geometry, relations), the most common slips, and a full side-by-side comparison with the other properties of equality (symmetric, transitive, addition, subtraction, multiplication, division, substitution).

Standard Form of Quadratic Equation — ax² + bx + c = 0

The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. This article walks through how to convert any quadratic to standard form, three worked examples at increasing difficulty, the most common conversion slips, and how standard form compares to vertex and factored form.

Zero Product Property — Definition, Formula, Examples

The zero product property states that if a product of factors equals zero, at least one factor must be zero. This article covers the formal statement, a four-step method, three worked examples at three difficulty tiers, an examples-by-equation-type reference table, a real-world word problem, the advantages and limits of the property, the common slips, and the cases where the property does not apply.

Set Builder Notation — Rules and Worked Examples

Set builder notation describes a set by stating the rule its elements obey — written as ${x \mid \text{condition on } x}$. This article covers the syntax, three worked examples, a side-by-side comparison with roster notation, and the small list of symbols you need to read it fluently.

Basics of Algebra — Variables, Expressions, Equations

The basics of algebra are five objects — variables, constants, expressions, equations, and inequalities — combined by the four arithmetic operations under one balance rule. This article walks through each foundational concept with three worked examples, the most common beginner errors, and the 9th-century origin of the word algebra itself.

Substitution Method for Systems of Equations

The substitution method solves a system of two linear equations by isolating one variable in one equation, plugging that expression into the other equation, then back-substituting. This article walks through the five steps, three worked examples at increasing difficulty, where students lose marks, and how substitution stacks up against elimination and graphing.

Cube Root of 27 — Simplified Form, Methods, and Proof

The cube root of 27 is exactly $\sqrt[3]{27} = 3$, because $3 \times 3 \times 3 = 27$. $27$ is a perfect cube, so the cube root is a clean integer — no decimal approximation needed. This article covers the value, the prime-factorisation method, where $\sqrt[3]{27}$ shows up, and the slips students make most often.

Cube Root of 343 — Simplified Form and How to Find It

The cube root of 343 is exactly $\sqrt[3]{343} = 7$, because $7 \times 7 \times 7 = 343$. $343$ is a perfect cube, so the cube root is a clean integer — no decimal approximation needed. This article covers the value, the prime-factorisation method, where $\sqrt[3]{343}$ shows up, and the slips students make most often.

Square Root of 61 — How to Find It

The square root of 61 is approximately $\sqrt{61} \approx 7.8102$ — irrational, non-terminating, and cannot be simplified into a cleaner radical form because $61$ is prime. This article covers the value in exact and decimal form, three methods to compute it, where $\sqrt{61}$ shows up, and the slips students make most often.

Square Root of 85 — Value, Method & How to Find

The square root of 85 is approximately $\sqrt{85} \approx 9.2195$ — irrational, non-terminating, and cannot be simplified into a cleaner radical form. This article covers the value in exact and decimal form, three methods to compute it, where $\sqrt{85}$ shows up, and the slips students make most often.

Vertical Line Test — How to Tell if a Graph Is a Function

The vertical line test says a graph represents a function if and only if no vertical line crosses it at more than one point. Draw a vertical line anywhere on the graph — if it hits the curve once (or zero times), keep going. If it hits twice, the relation is not a function. This article covers the rule, the pass-vs-fail table for common graphs, three worked examples, and the mistakes that quietly cost marks.

Exponential Form — Converting from Logarithmic & Radical

Exponential form writes repeated multiplication as a base raised to an exponent — for example $8 = 2^3$ instead of $2 \times 2 \times 2$. This article covers what exponential form is, how to convert to and from logarithmic and radical forms, the conversion-pattern table that captures every move on one page, three worked examples, and the mistakes that quietly cost marks.

Negative Exponents — Rules, Examples & How to Simplify

A negative exponent means take the reciprocal of the base, then apply the positive version of the exponent: $a^{-n} = \dfrac{1}{a^n}$. So $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$. This article covers the rule, the sign-pattern table that tells you what the answer looks like before you compute, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Types of Functions — Complete Classification with Examples

The types of functions are organised into three groups: by mapping (how inputs connect to outputs — one-one, onto, into, bijection), by degree (constant, linear, quadratic, cubic, polynomial), and by concept. This article walks through every type students meet from middle school to early college, with one-line examples and graphs where relevant.

Sigma Notation (Σ) - Summation Formula & Examples

Sigma notation uses the Greek capital letter $\Sigma$ to write a long sum compactly — for example, $\sum_{i=1}^{10} i$ means $1 + 2 + 3 + \dots + 10 = 55$. This article covers how to read sigma notation, the index conventions, the four core summation formulas, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

X Squared (x²) - Meaning, Graph, and Properties

x squared — written $x^2$ — means $x$ multiplied by itself. If $x = 3$, then $x^2 = 9$. This article covers what $x^2$ means, what its graph looks like (a parabola), the difference between $x^2$ and $2x$, the key properties of squaring, and a quick-reference power table for the small values you will use most.

Factored Form - Definition, Conversion, and Examples

The factored form of a quadratic writes it as a product of linear factors — for example $x^2 + 5x + 6$ becomes $(x+2)(x+3)$. This article covers what factored form is, how it compares to standard form and vertex form, how to convert between them, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Trinomial - Definition, Types, and Factoring Methods

A trinomial is a polynomial with exactly three unlike terms — most commonly written as $ax^2 + bx + c$. This article covers the definition, the three types of trinomials you will meet at school (quadratic, perfect square, cubic), the four factoring methods compared side by side, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Exponential Equations — Solving Methods, Formulas & Examples

An exponential equation is an equation where the variable sits in the exponent — like $2^x = 32$ or $3^{2x+1} = 81$. This article covers what makes an equation exponential, the three methods used to solve them (same base, logarithms, substitution), three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Fractional Exponents — Rules, Examples & How to Simplify

A fractional exponent is a root written as a power — $a^{1/n}$ means the $n$-th root of $a$, and $a^{m/n}$ means take the $n$-th root and raise it to the $m$-th power. This article covers the rules that govern fractional exponents, three worked examples at increasing difficulty, the three mistakes that quietly cost marks, and where rational powers turn up beyond the algebra classroom.

Cube Root of 64 - Value, Method, Examples

The cube root of 64 is exactly $4$, since $4^3 = 4 \times 4 \times 4 = 64$. Because $64$ is a perfect cube, its cube root is a whole number — no decimal, no approximation. The simplified radical form is $\sqrt[3]{64} = 4$.

Expression in Math - Definition, Types, Examples

A mathematical expression is a combination of numbers, variables, and operations that represents a value — but contains no equals sign. Examples: $3x + 5$, $2y^2 - 7$, $\sqrt{x + 1}$. An expression with an equals sign attached becomes an equation.

Determinant of a Matrix - Formula and Calculation

The determinant of a square matrix is a single number that captures the matrix's area-scaling factor (or volume-scaling in 3D), its invertibility (non-zero $\Leftrightarrow$ invertible), and the orientation it produces (positive or negative). For $2\times 2$: $\det = ad - bc$. For $3 \times 3$: cofactor expansion.

Adjoint of a Matrix - Formula, Steps, Examples

The adjoint of a matrix $A$ (also called adjugate) is the transpose of the cofactor matrix. It's the bridge that connects the matrix to its inverse: $A^{-1} = \dfrac{1}{\det(A)} \operatorname{adj}(A)$.

Inverse of 2x2 Matrix - Formula, Steps, Examples

The inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ is given by a clean formula: $A^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}$. The denominator $ad - bc$ is the determinant — if it's zero, the inverse doesn't exist.

Solving an Equation - Linear, Quadratic, Radical

Solving an equation means finding the value (or values) of the unknown that make both sides equal. The four-step universal process — simplify, isolate the variable, undo operations in reverse order, check — applies to every equation type.

How to Solve for x in Algebraic Equations

To solve for $x$ means to isolate $x$ on one side of an equation using legal moves. The method depends on the equation type — linear, quadratic, radical, rational, or exponential — but the underlying logic is always the same: undo operations in reverse order until $x$ stands alone.

Discriminant in Quadratic Equations - Formula & Examples

The discriminant $\Delta = b^2 - 4ac$ is the piece under the square root in the quadratic formula. Its sign tells you the nature of a quadratic's roots before you finish solving — two distinct real roots ($\Delta > 0$), one repeated real root ($\Delta = 0$), or two complex roots ($\Delta < 0$).

Properties of Logarithms - Laws, Formulas, Proofs

The properties of logarithms are seven identities that simplify logarithmic expressions: the product rule ($\log_b(xy) = \log_b x + \log_b y$), quotient rule ($\log_b(x/y) = \log_b x - \log_b y$), power rule ($\log_b(x^n) = n \log_b x$), change-of-base formula, and three special values ($\log_b 1 = 0$, $\log_b b = 1$, $\log_b b^n = n$).

Axis of Symmetry - Formula, Equation, Examples

The axis of symmetry of a parabola is the vertical line that splits the curve into two mirror-image halves. For a parabola in standard form $y = ax^2 + bx + c$, the equation of the axis of symmetry is $x = -\frac{b}{2a}$. In vertex form $y = a(x - h)^2 + k$, the axis is simply $x = h$.

Arithmetic Sequence - Formula, Definition, Examples

An arithmetic sequence (or arithmetic progression, AP) is an ordered list of numbers where the difference between consecutive terms is constant — that constant is called the common difference ($d$). The nth-term formula is $a_n = a_1 + (n-1)d$ and the sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$.

Exponent Rules - Laws of Exponents, Formulas, Examples

The eight core exponent rules simplify expressions involving powers: the product rule, the quotient rule, the power rule, and supporting rules for the power of a product, power of a quotient, zero exponent, negative exponent, and fractional exponent. This article gives the laws with worked examples, the historical origin, the entities involved, and the most common student mistakes.

Algebraic Expression - Parts, Types, and Examples

An algebraic expression is a mathematical phrase combining variables, constants, and operations — but with no equals sign. This article covers the formal definition, every part, the four types, simplification rules, the most common student mistakes, and where the entities live in modern math.

Square Root of 12 - Value, Simplification, Examples

The square root of 12 ($\sqrt{12}$) equals $2\sqrt{3}$ in simplest radical form, or approximately $3.464$ in decimal. This article covers the simplification step ($12 = 4 \times 3$, so $\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$), the long-division decimal computation, where $\sqrt{12}$ appears in geometry, and the common mistakes when simplifying radicals.

Square Root of 64 - Value, Calculation, Examples

The square root of 64 ($\sqrt{64}$) equals exactly $8$ — no decimal expansion, no irrationality, just a clean whole number. This article covers why $\sqrt{64} = 8$, three methods to compute it, where 64 shows up as a perfect square in computing and geometry, and the most common mistakes.

Square Root of 10 - Value, Calculation, Examples

The **square root of 10** ($\sqrt{10}$) is approximately $3.162$. This article gives the exact form, the decimal approximation to several places, the long-division method to compute it by hand, where $\sqrt{10}$ shows up in geometry (the diagonal of a $1 \times 3$ rectangle), and why it's irrational.

Linear Equations - Definition, Forms, and Graphs

A linear equation is an equation where the variables appear only to the first power - no squares, no cubes, no roots. The graph is always a straight line. This article covers the three standard forms, how to graph each, how to convert between them, and the most common solving mistakes.

Logarithm Rules - Product, Quotient, and Power

Three logarithm rules turn complicated arithmetic into simple arithmetic: the product rule ($\log_b(xy) = \log_b x + \log_b y$) turns multiplication into addition; the quotient rule turns division into subtraction; the power rule turns exponents into multiplication.

Multiplication of Matrices - Rules and Methods

The multiplication of matrices is defined only when the number of columns in the first matrix equals the number of rows in the second — and the result is computed row-by-column using dot products. This article covers the dimension rule, the step-by-step method, the four properties, worked examples in $2 \times 2$ and $3 \times 3$ cases and the most common mistakes.

Coefficient - Definition, Types, and Math Examples

A coefficient is the number that multiplies a variable in an algebraic expression — in $3x$, the coefficient is $3$. This article covers the formal definition, the four types of coefficients, how to identify them in any expression, the rule for variables with no visible number, and the most common mistakes when picking out the coefficient.

Sequences in Algebra - Types, Formulas, Examples

A sequence in algebra is an ordered list of numbers — called terms — that follows a specific rule. Sequences can be finite (ending) or infinite (continuing forever). The eight most-studied types are arithmetic (add a constant), geometric (multiply by a constant), harmonic (reciprocals of arithmetic), Fibonacci (each term = sum of previous two), triangular, square, cube, and quadratic.

Sets in Math - Definition, Types, and Symbols

A set in math is a well-defined collection of distinct objects, called elements, enclosed in curly braces — for example, $A = {2, 4, 6, 8}$. This article covers the formal definition, the 12 most common types of sets (finite, infinite, empty, singleton, equal, equivalent, subset, superset, universal, power, disjoint, overlapping), the symbols used to describe them ($\in$, $\subset$, $\cup$, $\cap$, $\varnothing$), the four core operations, and the mistakes that trip up most students.

Quadratic Equations - Formula, Solving, Examples

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$ where $a \neq 0$. This article covers the standard form, the three methods for solving (factoring, completing the square, and the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$).

Equation in Math - Definition, Types, and Parts

An equation in math is a statement that two expressions are equal, separated by an equals sign — for example, $3x + 5 = 14$. This article covers the formal definition, every part of an equation, the seven main types you'll meet in school, and the most common mistakes when solving them.

Algebra - Concepts, Topics, and Basic Formulas

Algebra is the branch of mathematics that uses letters and symbols — usually $x$, $y$, $z$ — to represent unknown numbers, and then works out what those numbers must be. This article covers the core concepts, the main topics, the basic formulas you'll use again and again.

Perfect Squares - List of Squares and Formula

Polynomials – Definition, Types, and Formulas