For a number n, its square root is x where x² = n. Denoted as $\sqrt{n}$ or $n^{1/2}$

For a number n, its cube root is x where x³ = n. Denoted as $\sqrt[3]{n}$ or $n^{1/3}$

√ for square root, $\sqrt[n]{x}$ for nth root where n is the index and x is the radicand

The expression x under the radical sign in $\sqrt[n]{x}$. The value we're finding the root of

The value n in $\sqrt[n]{x}$ indicating which root to take (square, cube, fourth, etc)

For an even root, the non-negative root out of all possible values. For odd roots, the real root

A number n = k² where k is an integer. Also called square number

A number n = k³ where k is an integer. Also called cubic number

Value x where $x^n = a$, denoted as $\sqrt[n]{a}$ or $a^{1/n}$

Mathematical expression containing one or more radicals $\sqrt[n]{x}$. Can include coefficients, variables, and operations

Converting radical to equivalent form with smallest possible radicand and rational coefficients outside

Expression containing radicals inside other radicals: $\sqrt{a + \sqrt{b}}$

An irrational root of a rational number. A radical that cannot be simplified to a rational number

Equation containing variable(s) under radical sign: $\sqrt{x} = a$ or $\sqrt{f(x)} = g(x)$

Root expressions written as powers: $x^{\frac{1}{n}} = \sqrt[n]{x}$ and $x^{\frac{m}{n}} = (\sqrt[n]{x})^m$

Multiplying numerator and denominator by radical term to eliminate radicals in denominator: $\frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$

A root that yields an irrational number - cannot be expressed as p/q where p,q are integers, q≠0

Methods to find approximate values of roots: Newton's method: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

For expression a + √b, its conjugate is a - √b. Product is a² - b. Used to rationalize denominators

$\sqrt[n]{a} = e^{\frac{\ln(a)}{n}}$ and $\sqrt[n]{a} = b \iff a = b^n$

nth root where n > 3: $\sqrt[n]{a}$ is value x where x^n = a

For negative real number -a, its square root is i√a where i = √(-1). Higher even roots also yield imaginary results

For positive numbers $b ≠ 1$ and $x > 0$, $log_b(x) = y$ means $b^y = x$. Written as: $\log_b(x) = y \iff b^y = x$

The positive number b ≠ 1 in logarithmic expression $\log_b(x)$ or exponential expression $b^x$

The power y in exponential form $b^y$ or the value of logarithm $\log_b(x)$ where $b^y = x$

Logarithm with base $e (≈ 2.71828...)$, written as $ln(x)$ or $\log_e(x)$. Inverse of exponential function $e^x$

Logarithm with base 10, written as $log(x)$ or $\log_{10}(x)$. Used for decimal representations

Logarithm with base 2, written as $\log_2(x)$. Used in computer science and information theory

The inverse logarithm function: if y = $\log_b(x)$, then $antilog_b(y) = x = b^y$

The integer part n of logarithm where $\log_{10}(x) = n + d$ and $0 ≤ d < 1$

The decimal part d of logarithm where $\log_{10}(x) = n + d$ and $0 ≤ d < 1$

Function $f(x) = \log_b(x)$ where $b > 0, b ≠ 1$. Inverse of exponential function $g(x) = b^x$

For complex $z = r(cos θ + i sin θ)$, $\ln(z) = \ln(r) + i(θ + 2πn)$ where n is integer

For integers $a, b, m$, find $x$ where $a^x ≡ b \pmod{m}$. Written as $\log_a(b) \pmod{m}$

Scale where values are spaced by powers of base b: positions proportional to $\log_b(x)$ rather than x

Equivalent expression of $\log_b(x) = y$ as $b^y = x$, showing inverse relationship between logarithms and exponents

Fundamental rules for manipulating logarithms with same base b:

Mathematical expression containing one or more logarithms, may include variables and other operations

Equation containing logarithmic expressions that must be solved for variable(s)

Inequality containing logarithmic expressions to be solved: $\log_b(x) < k$ or $\log_b(f(x)) > \log_b(g(x))$

For logarithmic function $f(x) = \log_b(x)$, vertical asymptote at $x = 0$

Plot of $y = \log_b(x)$ showing characteristic shape with vertical asymptote and continuous growth

$\log_a(x) = \frac{\log_c(x)}{\log_c(a)}$ for any base $c > 0, c ≠ 1$

Growth pattern where variable increases by additive constant when input is multiplied by constant: $f(cx) = f(x) + k$

Converting data by taking logarithm: $y = \log_b(x)$ to linearize relationships or normalize distributions

An expression consisting of variables, coefficients, and non-negative integer exponents combined using arithmetic operations.

A numerical factor that multiplies a variable in a polynomial.

The coefficient of the term with the highest degree in a polynomial.

The constant term in a polynomial with no variable attached.

The highest exponent of the variable in a polynomial.

A polynomial whose leading coefficient is 1.

A polynomial where all coefficients are zero.

A function that always evaluates to zero for any input.

The degree of the zero polynomial, which is not defined.

An informal term sometimes used to describe the degree of the zero polynomial.

An equation that sets a polynomial equal to another expression, typically zero.

The sum of two or more polynomials by adding corresponding terms.

The difference of two polynomials by subtracting corresponding terms.

The product of two polynomials by distributing terms.

The process of dividing one polynomial by another, yielding a quotient and remainder.

The polynomial being divided in a division operation.

The polynomial by which another polynomial is divided.

The result of polynomial division before considering the remainder.

The leftover polynomial after division that cannot be further divided by the divisor.

A function expressed as the ratio of two polynomials.

A theorem stating that the remainder when a polynomial P(x) is divided by (x - a) is P(a).

A value of the variable that makes the polynomial equal to zero.

The sum of all coefficients of a polynomial is found by evaluating it at x = 1.

Expressing a polynomial as a product of simpler polynomials.

A formula used to find the roots of a quadratic polynomial.

Non-real solutions of a polynomial equation, often involving imaginary numbers.

The number of times a particular root appears in the factorization of a polynomial.

A theorem stating that every non-constant polynomial has at least one complex root.

A principle ensuring the existence of at least one solution for a given polynomial equation.

The decomposition of a polynomial into a product of lower-degree polynomials.

The derivative of a polynomial function, obtained by differentiating each term.

A root of a polynomial whose multiplicity is greater than one.

Expressing a polynomial as a product of linear factors corresponding to its roots.

A strategy for guessing rational roots of polynomials using integer coefficients.

A polynomial where all coefficients are whole numbers.

A theorem that provides a possible set of rational roots for a polynomial with integer coefficients.

A theorem stating that polynomials can be factored uniquely over specific number systems.

A division algorithm for polynomials similar to numerical long division.

A root of a polynomial that appears more than once.

A mathematical field containing a finite number of elements.

A system of arithmetic where numbers wrap around after reaching a fixed modulus.

The process of breaking a polynomial into irreducible factors within a finite field.

A number that is a square modulo a given prime number.

The solutions to a polynomial equation within a finite field.

A theorem stating that a polynomial of degree n over a finite field has at most n roots in that field.

A technique for solving polynomials by directly substituting values.

A modified version of the quadratic formula adapted for finite fields.

A root of a polynomial in a finite field that has higher multiplicity.

A set of equations relating the coefficients of a polynomial to sums and products of its roots.

The sum of the roots of a polynomial, given by Vieta’s formulas.

The product of the roots of a polynomial, also given by Vieta’s formulas.

A method for finding unknown coefficients by equating polynomials.

The number of times a particular root appears in the factorization of a polynomial.

The process of expanding a factored polynomial into standard form.

A compact way to represent summation, often used in polynomial expressions.

The process of factoring a polynomial using its known roots.

Relationships between the roots and coefficients of quadratic and higher-degree polynomials.

The power to which a base is raised in a mathematical expression.

The number that is raised to the power of the exponent.

The result of raising a base to an exponent.

A function of the form f(x) = a^x, where a > 0 and a ≠ 1.

A process that increases proportionally to its current value, modeled by an exponential function.

A process that decreases proportionally to its current value, modeled by an exponential function.

A method of writing numbers using powers of 10.

Indicates the reciprocal of the base raised to the corresponding positive exponent (a^(-n) = 1/a^n).

Any nonzero base raised to the power of zero equals 1 (a^0 = 1).

Represents a root, where a^(1/n) = √[n]{a}.

An exponent that is a whole number.

An equation in which variables appear in exponents.

An inequality involving exponential expressions.

A mathematical expansion of e^x as a sum of terms.

The mathematical operation of raising one quantity to the power of another.

Rules governing exponentiation, such as the product, quotient, and power rules.

The graph of an exponential function, showing rapid increase or decrease.

The exponential function with base e, f(x) = e^x.

A financial formula based on exponential growth, A = P(1 + r/n)^(nt).

A shorthand way to write repeated multiplication of the same factor.

For a number n, its square root is x where x² = n. Denoted as $\sqrt{n}$ or $n^{1/2}$

For a number n, its cube root is x where x³ = n. Denoted as $\sqrt[3]{n}$ or $n^{1/3}$

√ for square root, $\sqrt[n]{x}$ for nth root where n is the index and x is the radicand

The expression x under the radical sign in $\sqrt[n]{x}$. The value we're finding the root of

The value n in $\sqrt[n]{x}$ indicating which root to take (square, cube, fourth, etc)

For an even root, the non-negative root out of all possible values. For odd roots, the real root

A number n = k² where k is an integer. Also called square number

A number n = k³ where k is an integer. Also called cubic number

Value x where $x^n = a$, denoted as $\sqrt[n]{a}$ or $a^{1/n}$

Mathematical expression containing one or more radicals $\sqrt[n]{x}$. Can include coefficients, variables, and operations

Converting radical to equivalent form with smallest possible radicand and rational coefficients outside

Expression containing radicals inside other radicals: $\sqrt{a + \sqrt{b}}$

An irrational root of a rational number. A radical that cannot be simplified to a rational number

Equation containing variable(s) under radical sign: $\sqrt{x} = a$ or $\sqrt{f(x)} = g(x)$

Root expressions written as powers: $x^{\frac{1}{n}} = \sqrt[n]{x}$ and $x^{\frac{m}{n}} = (\sqrt[n]{x})^m$

Multiplying numerator and denominator by radical term to eliminate radicals in denominator: $\frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$

A root that yields an irrational number - cannot be expressed as p/q where p,q are integers, q≠0

Methods to find approximate values of roots: Newton's method: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

For expression a + √b, its conjugate is a - √b. Product is a² - b. Used to rationalize denominators

$\sqrt[n]{a} = e^{\frac{\ln(a)}{n}}$ and $\sqrt[n]{a} = b \iff a = b^n$

nth root where n > 3: $\sqrt[n]{a}$ is value x where x^n = a

For negative real number -a, its square root is i√a where i = √(-1). Higher even roots also yield imaginary results

For positive numbers $b ≠ 1$ and $x > 0$, $log_b(x) = y$ means $b^y = x$. Written as: $\log_b(x) = y \iff b^y = x$

The positive number b ≠ 1 in logarithmic expression $\log_b(x)$ or exponential expression $b^x$

The power y in exponential form $b^y$ or the value of logarithm $\log_b(x)$ where $b^y = x$

Logarithm with base $e (≈ 2.71828...)$, written as $ln(x)$ or $\log_e(x)$. Inverse of exponential function $e^x$

Logarithm with base 10, written as $log(x)$ or $\log_{10}(x)$. Used for decimal representations

Logarithm with base 2, written as $\log_2(x)$. Used in computer science and information theory

The inverse logarithm function: if y = $\log_b(x)$, then $antilog_b(y) = x = b^y$

The integer part n of logarithm where $\log_{10}(x) = n + d$ and $0 ≤ d < 1$

The decimal part d of logarithm where $\log_{10}(x) = n + d$ and $0 ≤ d < 1$

Function $f(x) = \log_b(x)$ where $b > 0, b ≠ 1$. Inverse of exponential function $g(x) = b^x$

For complex $z = r(cos θ + i sin θ)$, $\ln(z) = \ln(r) + i(θ + 2πn)$ where n is integer

For integers $a, b, m$, find $x$ where $a^x ≡ b \pmod{m}$. Written as $\log_a(b) \pmod{m}$

Scale where values are spaced by powers of base b: positions proportional to $\log_b(x)$ rather than x

Equivalent expression of $\log_b(x) = y$ as $b^y = x$, showing inverse relationship between logarithms and exponents

Fundamental rules for manipulating logarithms with same base b:

Mathematical expression containing one or more logarithms, may include variables and other operations

Equation containing logarithmic expressions that must be solved for variable(s)

Inequality containing logarithmic expressions to be solved: $\log_b(x) < k$ or $\log_b(f(x)) > \log_b(g(x))$

For logarithmic function $f(x) = \log_b(x)$, vertical asymptote at $x = 0$

Plot of $y = \log_b(x)$ showing characteristic shape with vertical asymptote and continuous growth

$\log_a(x) = \frac{\log_c(x)}{\log_c(a)}$ for any base $c > 0, c ≠ 1$

Growth pattern where variable increases by additive constant when input is multiplied by constant: $f(cx) = f(x) + k$

Converting data by taking logarithm: $y = \log_b(x)$ to linearize relationships or normalize distributions

An expression consisting of variables, coefficients, and non-negative integer exponents combined using arithmetic operations.

A numerical factor that multiplies a variable in a polynomial.

The coefficient of the term with the highest degree in a polynomial.

The constant term in a polynomial with no variable attached.

The highest exponent of the variable in a polynomial.

A polynomial whose leading coefficient is 1.

A polynomial where all coefficients are zero.

A function that always evaluates to zero for any input.

The degree of the zero polynomial, which is not defined.

An informal term sometimes used to describe the degree of the zero polynomial.

An equation that sets a polynomial equal to another expression, typically zero.

The sum of two or more polynomials by adding corresponding terms.

The difference of two polynomials by subtracting corresponding terms.

The product of two polynomials by distributing terms.

The process of dividing one polynomial by another, yielding a quotient and remainder.

The polynomial being divided in a division operation.

The polynomial by which another polynomial is divided.

The result of polynomial division before considering the remainder.

The leftover polynomial after division that cannot be further divided by the divisor.

A function expressed as the ratio of two polynomials.

A theorem stating that the remainder when a polynomial P(x) is divided by (x - a) is P(a).

A value of the variable that makes the polynomial equal to zero.

The sum of all coefficients of a polynomial is found by evaluating it at x = 1.

Expressing a polynomial as a product of simpler polynomials.

A formula used to find the roots of a quadratic polynomial.

Non-real solutions of a polynomial equation, often involving imaginary numbers.

The number of times a particular root appears in the factorization of a polynomial.

A theorem stating that every non-constant polynomial has at least one complex root.

A principle ensuring the existence of at least one solution for a given polynomial equation.

The decomposition of a polynomial into a product of lower-degree polynomials.

The derivative of a polynomial function, obtained by differentiating each term.

A root of a polynomial whose multiplicity is greater than one.

Expressing a polynomial as a product of linear factors corresponding to its roots.

A strategy for guessing rational roots of polynomials using integer coefficients.

A polynomial where all coefficients are whole numbers.

A theorem that provides a possible set of rational roots for a polynomial with integer coefficients.

A theorem stating that polynomials can be factored uniquely over specific number systems.

A division algorithm for polynomials similar to numerical long division.

A root of a polynomial that appears more than once.

A mathematical field containing a finite number of elements.

A system of arithmetic where numbers wrap around after reaching a fixed modulus.

The process of breaking a polynomial into irreducible factors within a finite field.

A number that is a square modulo a given prime number.

The solutions to a polynomial equation within a finite field.

A theorem stating that a polynomial of degree n over a finite field has at most n roots in that field.

A technique for solving polynomials by directly substituting values.

A modified version of the quadratic formula adapted for finite fields.

A root of a polynomial in a finite field that has higher multiplicity.

A set of equations relating the coefficients of a polynomial to sums and products of its roots.

The sum of the roots of a polynomial, given by Vieta’s formulas.

The product of the roots of a polynomial, also given by Vieta’s formulas.

A method for finding unknown coefficients by equating polynomials.

The number of times a particular root appears in the factorization of a polynomial.

The process of expanding a factored polynomial into standard form.

A compact way to represent summation, often used in polynomial expressions.

The process of factoring a polynomial using its known roots.

Relationships between the roots and coefficients of quadratic and higher-degree polynomials.

The power to which a base is raised in a mathematical expression.

The number that is raised to the power of the exponent.

The result of raising a base to an exponent.

A function of the form f(x) = a^x, where a > 0 and a ≠ 1.

A process that increases proportionally to its current value, modeled by an exponential function.

A process that decreases proportionally to its current value, modeled by an exponential function.

A method of writing numbers using powers of 10.

Indicates the reciprocal of the base raised to the corresponding positive exponent (a^(-n) = 1/a^n).

Any nonzero base raised to the power of zero equals 1 (a^0 = 1).

Represents a root, where a^(1/n) = √[n]{a}.

An exponent that is a whole number.

An equation in which variables appear in exponents.

An inequality involving exponential expressions.

A mathematical expansion of e^x as a sum of terms.

The mathematical operation of raising one quantity to the power of another.

Rules governing exponentiation, such as the product, quotient, and power rules.

The graph of an exponential function, showing rapid increase or decrease.

The exponential function with base e, f(x) = e^x.

A financial formula based on exponential growth, A = P(1 + r/n)^(nt).

A shorthand way to write repeated multiplication of the same factor.