Comprehensive Study Notes on Proportionality, Rule of Three, Functions, Radicals, Rational Exponents, Power Equations, Irrational Operations, and Notable Products

Proportionality and Ratios

A ratio represents a quantitative comparison between two real numbers $a$ and $b$, where $b \neq 0$, expressed in quotient form as $\frac{a}{b}$ or $a : b$. In this mathematical relationship, $a$ is referred to as the antecedent and $b$ as the consequent. A proportion is defined as an equality between two ratios, represented algebraically as $\frac{a}{b} = \frac{c}{d}$, where $b \neq 0$ and $d \neq 0$. The values $a$ and $d$ are called the extremes, while $b$ and $c$ are called the means. The fundamental property of proportions asserts that the product of the extremes is equal to the product of the means, which yields the fundamental equation $a \times d = b \times c$.

Two variable quantities are directly proportional if an increase in one quantity causes a proportional increase in the other, maintaining a constant quotient $k$. Formally, two variable quantities $x$ and $y$ are directly proportional if there exists a non-zero constant $k$ such that $\frac{y}{x} = k$, or equivalently $y = k \times x$. Conversely, two quantities are inversely proportional if an increase in one quantity results in a proportional decrease in the other, keeping their product constant. Formally, two variable quantities $x$ and $y$ are inversely proportional if there exists a non-zero constant $k$ such that $x \times y = k$, or equivalently $y = \frac{k}{x}$.

Simple and Compound Rule of Three

The Simple Rule of Three is a practical mathematical algorithm used to compute an unknown value $x$ from three known values when dealing with two proportional quantities. When the quantities are directly proportional, the relationship is established by setting up equal ratios $\frac{a_1}{a_2} = \frac{b_1}{x}$, which yields the solution $x = \frac{a_2 \times b_1}{a_1}$ via cross-multiplication. When the quantities are inversely proportional, the relationship requires multiplying across corresponding pairs rather than cross-multiplying, leading to the equality $a_1 \times b_1 = a_2 \times x$, which yields the solution $x = \frac{a_1 \times b_1}{a_2}$.

The Compound Rule of Three extends this methodology to scenarios involving three or more interdependent quantities. To solve a compound rule of three problem, one must first identify the target quantity containing the unknown variable $x$. Then, each secondary quantity is compared independently to the target quantity to determine whether it shares a directly proportional or inversely proportional relationship, holding all other factors constant. Inverted ratios are constructed for inversely proportional quantities, while direct ratios are maintained for directly proportional ones. The final proportion equates the ratio of the target quantity to the product of all other adjusted ratio fractions, allowing for precise isolation and computation of $x$.

Mathematical Functions and Affine Functions

A mathematical function $f$ from a set $A$ to a set $B$, denoted as $f: A \rightarrow B$, is a binary relation that assigns to each element $x \in A$ exactly one unique element $y \in B$. The set $A$ is designated as the domain of the function, representing all valid input values. The set $B$ is the codomain, representing the set of all potential output values. The image set or range, denoted as $\text{Im}(f)$, is the subset of $B$ containing all actual output values generated by applying $f$ to the elements of $A$. Functions are formally expressed through an algebraic law of formation, written in the form $y = f(x)$, which specifies the exact mathematical operations performed on the independent variable $x$ to obtain the dependent variable $y$.

An affine function, also known as a linear function, is a specific class of function defined by the law of formation $f(x) = a \times x + b$, where $a$ and $b$ are real numbers and $a \neq 0$. The parameter $a$ is called the slope or angular coefficient, which determines the rate of change and the inclination of the linear graph. The parameter $b$ is called the y-intercept or linear coefficient, representing the precise point where the line intersects the vertical axis at $(0, b)$. When $a > 0$, the function is strictly increasing; when $a < 0$, the function is strictly decreasing; when $a = 0$, the expression simplifies to a constant function $f(x) = b$. The root or zero of an affine function is the unique input value $x$ for which $f(x) = 0$, obtained algebraically by solving $a \times x + b = 0$, which yields $x = -\frac{b}{a}$.

Radication and Properties of Radicals

Radication is the inverse mathematical operation of exponentiation. Given a real number $a$ and a positive integer $n$, the $n$-th root of $a$ is a number $b$ such that $b^n = a$. This operation is expressed using the radical notation $\sqrt[n]{a} = b$. The components of this expression are the radical symbol $\sqrt{}$, the index $n$ which specifies the root degree, the radicand $a$ positioned underneath the radical symbol, and the resulting root $b$. When the index $n$ is equal to $2$, it is standard convention to omit the index and write simply $\sqrt{a}$ to denote the principal square root.

Radical operations obey fundamental algebraic properties that allow for simplification and transformation. The product property of radicals establishes that the root of a product equals the product of the individual roots, written as $\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}$ for non-negative real numbers $a$ and $b$. The quotient property of radicals asserts that the root of a quotient equals the quotient of the individual roots, written as $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ where $b > 0$. Additional operational properties include raising a radical to a power, $(\sqrt[n]{a})^m = \sqrt[n]{a^m}$, and taking the root of a radical, $\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \times n]{a}$. Furthermore, the index and exponent can be simplified by dividing both by a common non-zero factor $k$, such that $\sqrt[n \times k]{a^{m \times k}} = \sqrt[n]{a^m}$.

Exponentiation with Rational and Radical Exponents

Exponentiation can be extended beyond integer exponents to encompass rational exponents, establishing a direct relationship between powers and radical expressions. For any real base $a \ge 0$ and rational exponent $\frac{m}{n}$, where $m$ is an integer and $n$ is a positive integer, the power with a fractional exponent is defined as $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Alternatively, this can be expressed as $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$. In this conversion, the numerator $m$ of the fractional exponent represents the power to which the base is raised, while the denominator $n$ becomes the index of the radical root.

All standard laws of exponents remain valid when operating with rational and radical exponents. The product of powers with the same base requires adding the exponents, $a^r \times a^s = a^{r + s}$. The quotient of powers with the same base requires subtracting the exponents, $\frac{a^r}{a^s} = a^{r - s}$. Raising a power to another exponent requires multiplying the exponents, $(a^r)^s = a^{r \times s}$. The power of a product distributes across individual factors, $(a \times b)^r = a^r \times b^r$, and the power of a quotient distributes across the numerator and denominator, $\left(\frac{a}{b}\right)^r = \frac{a^r}{b^r}$. Negative rational exponents follow the identity $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}$.

Power Equations and Exponential Relationships

A power equation in a single variable is an algebraic equation where the variable is raised to a constant exponent, generally expressed in the form $x^a = b$. Solving for the unknown variable $x$ depends on whether the exponent $a$ is even or odd, as well as the sign of the constant $b$. When $a$ is a positive odd integer, the equation has a single real solution given by $x = \sqrt[a]{b}$, which preserves the algebraic sign of $b$. When $a$ is a positive even integer and $b > 0$, the equation yields two distinct real solutions, $x = \sqrt[a]{b}$ and $x = -\sqrt[a]{b}$, conventionally written together as $x = \pm \sqrt[a]{b}$. If $a$ is even and $b < 0$, there are no real solutions within the set of real numbers $\mathbb{R}$.

When equations contain fractional or radical exponents, such as $x^{\frac{m}{n}} = b$, the isolation of $x$ is performed by applying the inverse exponent to both sides of the equation. Raising both sides to the power of $\frac{n}{m}$ yields $(x^{\frac{m}{n}})^{\frac{n}{m}} = b^{\frac{n}{m}}$, which simplifies to $x = b^{\frac{n}{m}} = \sqrt[m]{b^n}$, provided all real domain restrictions are satisfied. Care must be exercised during this process to verify solutions and eliminate any extraneous roots introduced when raising equations to even powers.

Operations with Irrational Numbers and Rationalization

Irrational numbers are real numbers that cannot be expressed as a simple fraction $\frac{a}{b}$ of two integers, possessing non-repeating, infinite decimal expansions. Operations involving irrational terms, specifically those expressed as radicals, require specific algebraic rules. Addition and subtraction can only be performed between like radicals—terms that share both the exact same index and radicand. When combining like radicals, the numerical coefficients are added or subtracted while maintaining the shared radical component, represented algebraically as $c \times \sqrt[n]{a} + d \times \sqrt[n]{a} = (c + d) \times \sqrt[n]{a}$.

Multiplication and division of irrational radicals with identical indices are performed by applying the product and quotient properties directly under a single radical sign. When indices differ, terms must first be transformed into equivalent radicals with a common index by finding the least common multiple of the indices. Rationalization of the denominator is the algebraic process of eliminating radicals from the denominator of a fraction. For a monomial denominator of the form $\frac{c}{\sqrt{a}}$, rationalization is accomplished by multiplying both the numerator and denominator by $\sqrt{a}$, yielding $\frac{c \times \sqrt{a}}{a}$. For a binomial denominator containing square roots, such as $\frac{c}{\sqrt{a} + \sqrt{b}}$, rationalization requires multiplying the numerator and denominator by the algebraic conjugate $\sqrt{a} - \sqrt{b}$, producing $\frac{c \times (\sqrt{a} - \sqrt{b})}{a - b}$.

Notable Products and the Square of a Sum

Notable products are algebraic identities that occur frequently in polynomial expansions and factorizations, allowing for immediate simplification without performing explicit polynomial multiplication step-by-step. The square of a sum is one of the foundational notable products, represented by the algebraic identity $(a + b)^2 = a^2 + 2 \times a \times b + b^2$. This identity states that the square of the sum of two terms is equal to the square of the first term, plus twice the product of the first and second terms, plus the square of the second term.

The algebraic proof of the square of a sum identity is obtained by applying the distributive property of multiplication over addition to the expanded expression $(a + b)(a + b)$. Multiplying each term yields $a \times a + a \times b + b \times a + b \times b$. Applying the commutative property of multiplication allows combining the like middle terms $a \times b + b \times a = 2 \times a \times b$, resulting in the final expanded form $a^2 + 2 \times a \times b + b^2$. Geometrically, this identity can be visualized as a large square with side length $a + b$ and total area $(a + b)^2$, partitioned into four internal sub-regions: one square of area $a^2$, another square of area $b^2$, and two identical rectangles each having an area of $a \times b$.