Glossary
Conversion (Radical to Exponential/vice versa)
The ability to rewrite expressions from radical form (ⁿ√a) to exponential form (a^(1/n)) and vice versa, which is crucial for simplification and solving.
Example:
The conversion of the radical ∛(x+5) to exponential form is (x+5)^(1/3).
Denominator (of rational exponent)
The bottom number in a fractional exponent, which represents the root to be taken (and corresponds to the index of the radical).
Example:
In y^(2/5), the denominator 5 indicates that the fifth root of y is taken.
Extraneous Roots
Solutions obtained during the solving process of radical equations that do not satisfy the original equation when checked.
Example:
When solving √(x+2) = x, squaring both sides might yield x=2 and x=-1; however, x=-1 is an extraneous root because it doesn't work in the original equation.
Index
The small number placed outside the radical symbol that indicates which root is being taken (e.g., 3 for a cube root, or an implied 2 for a square root).
Example:
In the expression ∛27, the number 3 is the index, telling us to find the cube root of 27.
Isolate (radical term)
The first step in solving radical equations, where the radical expression is moved to one side of the equation by itself.
Example:
Before squaring both sides of √x + 3 = 7, you must isolate the radical term by subtracting 3 from both sides, getting √x = 4.
Numerator (of rational exponent)
The top number in a fractional exponent, which represents the power to which the base is raised.
Example:
In x^(3/4), the numerator 3 indicates that x is raised to the third power.
Power Rule (Radicals)
A property stating that raising a radical to a power is equivalent to taking the root of the base raised to that power, i.e., (ⁿ√a)ᵐ = ⁿ√(aᵐ).
Example:
According to the Power Rule (Radicals), (√x)⁴ simplifies to √(x⁴) = x².
Power Rule (Rational Exponents)
A property stating that when raising an expression with a rational exponent to another power, you multiply the exponents.
Example:
Applying the Power Rule (Rational Exponents), (z^(2/3))^(3/2) simplifies to z^((2/3)*(3/2)) = z¹ = z.
Product Rule (Radicals)
A property stating that the root of a product is equal to the product of the roots, i.e., √(a * b) = √a * √b.
Example:
Using the Product Rule (Radicals), √72 can be simplified as √(36 * 2) = √36 * √2 = 6√2.
Product Rule (Rational Exponents)
A property stating that when multiplying expressions with the same base, you add their rational exponents.
Example:
Using the Product Rule (Rational Exponents), x^(1/2) * x^(1/3) becomes x^((1/2)+(1/3)) = x^(5/6).
Quotient Rule (Radicals)
A property stating that the root of a quotient is equal to the quotient of the roots, i.e., √(a / b) = √a / √b.
Example:
The Quotient Rule (Radicals) allows us to simplify √(49/16) as √49 / √16 = 7/4.
Quotient Rule (Rational Exponents)
A property stating that when dividing expressions with the same base, you subtract their rational exponents.
Example:
By the Quotient Rule (Rational Exponents), y^(3/4) / y^(1/4) simplifies to y^((3/4)-(1/4)) = y^(2/4) = y^(1/2).
Radicals
Mathematical expressions involving roots, such as square roots (√) or cube roots (∛), used to find a number that, when multiplied by itself a certain number of times, equals a given value.
Example:
To find the side length of a square with an area of 25, you would calculate the radical √25, which is 5.
Rational Exponents
Exponents expressed as fractions (a/b), where the numerator indicates the power and the denominator indicates the root.
Example:
The expression 8^(2/3) is a rational exponent that means the cube root of 8, squared, resulting in (∛8)² = 2² = 4.
Simplifying Radicals
The process of rewriting a radical expression so that the radicand contains no perfect square (or cube, etc.) factors other than 1.
Example:
To perform Simplifying Radicals, √50 can be broken down into √(25 * 2) = 5√2.