Enhance Your Learning with Radical Expressions and Equations Flash Cards for quick understanding
A symbol (√) used to represent the square root or other roots of a number.
An expression that contains a radical (√) symbol.
The number written above the radical symbol (√) that indicates the root being taken.
The number or expression inside the radical symbol (√).
To simplify a radical expression means to rewrite it in its simplest form by removing any perfect square factors from the radicand.
The square root (√) of a number is a value that, when multiplied by itself, gives the original number.
The cube root (∛) of a number is a value that, when multiplied by itself three times, gives the original number.
The fourth root (∜) of a number is a value that, when multiplied by itself four times, gives the original number.
The nth root (√ₙ) of a number is a value that, when multiplied by itself n times, gives the original number.
The process of eliminating radicals from the denominator of a fraction by multiplying both the numerator and denominator by a suitable expression.
To add or subtract radical expressions, combine like terms by adding or subtracting the coefficients of the like radicals.
To multiply radical expressions, multiply the coefficients and multiply the radicands.
To divide radical expressions, divide the coefficients and divide the radicands.
The conjugate of a binomial is obtained by changing the sign between the terms.
To solve radical equations, isolate the radical term and then raise both sides of the equation to the appropriate power to eliminate the radical.
An extraneous solution is a solution that does not satisfy the original equation.
An equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
An equation that contains a radical (√) symbol.
A radical equation has one solution if the equation simplifies to a true statement after isolating the radical term and squaring both sides.
A radical equation has no solution if the equation simplifies to a false statement after isolating the radical term and squaring both sides.
A relationship between two variables in which their ratio is constant.
A relationship between two variables in which their product is constant.
A relationship between three or more variables in which one variable varies directly with the product of the other variables.
A function that contains a radical (√) symbol.
The set of all possible input values (x-values) of a function.
The set of all possible output values (y-values) of a function.
To graph radical functions, plot points by substituting different x-values into the function and then connect the points with a smooth curve.
A radical function with an even index (2, 4, 6, etc.) that is symmetric with respect to the y-axis.
A radical function with an odd index (3, 5, 7, etc.) that is symmetric with respect to the origin.
A transformation of a function that shifts the graph vertically up or down.
A transformation of a function that shifts the graph horizontally left or right.
A transformation of a function that vertically stretches or compresses the graph.
A transformation of a function that horizontally stretches or compresses the graph.
A transformation of a function that reflects the graph across a line.
A function of the form f(x) = |x|, where f(x) represents the distance between x and 0 on the number line.
A function that is defined by different equations or expressions for different intervals of the domain.
An inequality that contains a radical (√) symbol.
To solve radical inequalities, isolate the radical term and then square both sides of the inequality. Remember to check for extraneous solutions.
An inequality that consists of two or more inequalities joined by the words 'and' or 'or'.
The set of values that satisfy both inequalities in a compound inequality joined by 'and'.
The set of values that satisfy at least one of the inequalities in a compound inequality joined by 'or'.
An inequality that contains an absolute value expression.
To solve absolute value inequalities, isolate the absolute value expression and then split the inequality into two cases, one with a positive and one with a negative.
A way to represent the solution set of an inequality using intervals on the number line.
A way to represent the solution set of an inequality using set notation.
A function of the form f(x) = a^x, where a is a positive constant and x is a real number.
A type of exponential function where the base (a) is greater than 1, resulting in a graph that increases rapidly.
A type of exponential function where the base (a) is between 0 and 1, resulting in a graph that decreases rapidly.
Interest that is calculated on the initial principal and any accumulated interest from previous periods.
Interest that is compounded continuously, resulting in a formula of the form A = P * e^(rt), where A is the final amount, P is the principal, r is the interest rate, and t is the time in years.
The time it takes for half of a substance to decay or disintegrate.
The inverse operation of exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to obtain that number.
A logarithm with base 10, often written as log(x).
A logarithm with base e, where e is the mathematical constant approximately equal to 2.71828, often written as ln(x).
A function of the form f(x) = logₐ(x), where a is the base and x is the argument.
Various rules and properties that can be used to simplify and manipulate logarithmic expressions.
A formula used to evaluate logarithms with bases other than 10 or e, often written as logₐ(x) = log(x) / log(a).
An equation that contains an exponential function.
To solve exponential equations, take the logarithm of both sides of the equation and use the properties of logarithms to simplify and solve for the variable.
An equation that contains a logarithmic function.
To solve logarithmic equations, rewrite the equation in exponential form and solve for the variable.
A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years.
A = P(1/2)^(t/h), where A is the final amount, P is the initial amount, t is the time elapsed, and h is the half-life.
logₐ(xy) = logₐ(x) + logₐ(y), where a is the base and x and y are positive numbers.
logₐ(x/y) = logₐ(x) - logₐ(y), where a is the base and x and y are positive numbers.
logₐ(x^r) = r * logₐ(x), where a is the base, x is a positive number, and r is a real number.
logₐ(x) = log(x) / log(a), where a is the base and x is a positive number.
a^x * a^y = a^(x + y), where a is the base and x and y are real numbers.
a^x / a^y = a^(x - y), where a is the base and x and y are real numbers.
(a^x)^y = a^(xy), where a is the base and x and y are real numbers.
a^0 = 1, where a is the base.