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How do you simplify a radical expression?
Look for perfect square, cube, or higher order factors within the radical and break them down. Example: √18 = √(9 * 2) = 3√2
Describe the first step in solving an equation with radicals.
Isolate the radical term on one side of the equation.
What must you always do when solving radical equations?
Check your solution(s) in the original equation to avoid extraneous roots.
How do you eliminate a radical in an equation?
Raise both sides of the equation to the power of the index of the radical.
Explain how to convert from radical to exponential form.
The index of the radical becomes the denominator of the fractional exponent: ⁿ√a = a^(1/n)
Explain how to convert from exponential to radical form.
The denominator of the fractional exponent becomes the index of the radical: a^(1/n) = ⁿ√a
Simplify: $\sqrt[3]{54x^6y^4}$
\$\sqrt[3]{54x^6y^4} = \sqrt[3]{27 \cdot 2 \cdot (x^2)^3 \cdot y^3 \cdot y} = 3x^2y\sqrt[3]{2y}$
Solve for x: $(2x - 1)^{3/2} = 27$
$(2x - 1)^{3/2} = 27 \implies (2x-1) = 27^{2/3} = 9 \implies 2x = 10 \implies x = 5$
How do you find the domain of $$f(x) = (x^2 - 4)^{1/2}$$?
Set $$x^2 - 4 \ge 0$$, which implies $$x^2 \ge 4$$. Therefore, $$x \le -2$$ or $$x \ge 2$$
Evaluate $$f(x) = (x^2 - 4)^{1/2}$$ at $$x=5$$
$$f(5) = (5^2 - 4)^{1/2} = (25-4)^{1/2} = \sqrt{21}$$
Define 'radical'.
A radical involves roots like square roots (√), cube roots (∛), and beyond.
What is the 'index' of a radical?
The index tells you what kind of root you're dealing with (e.g., 3 in ∛).
Define 'rational exponent'.
An exponent that is a fraction (like a/b). The numerator is the power, and the denominator is the root.
State the conversion formula between radical and exponential forms.
ⁿ√a = a^(1/n)
State the Product Rule for Radicals.
√(a * b) = √a * √b
State the Quotient Rule for Radicals.
√(a / b) = √a / √b
State the Power Rule for Radicals.
(√a)^n = √[n](a)
State the Product Rule for Rational Exponents.
a^(m/n) * a^(p/q) = a^((mq + np) / nq)
State the Quotient Rule for Rational Exponents.
a^(m/n) / a^(p/q) = a^((mq - np) / nq)
State the Power Rule for Rational Exponents.
(a^(m/n))^(p/q) = a^((mp) / (nq))