Radicals and Rational Exponents flashcards

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In Nth Roots and Rational Exponents, what does principal even roots mean?Nth Roots and Rational Exponents: The defined nonnegative output of an even root in the reals.
Is √(x²) always x?Nth Roots and Rational Exponents: No, it is |x| for real x.
In Nth Roots and Rational Exponents, what does odd roots mean?Nth Roots and Rational Exponents: A real nth root for an odd index.
Why can ∛(−8) equal −2?Nth Roots and Rational Exponents: Because (−2)³=−8.
In Nth Roots and Rational Exponents, what does rational exponents mean?Nth Roots and Rational Exponents: A fractional-power notation for roots and powers.
Why state the real domain?Nth Roots and Rational Exponents: An even root of a negative base is not real.
In Simplifying Radical Expressions, what does perfect-power factors mean?Simplifying Radical Expressions: Removing known perfect powers from a root.
Why is √(x²)=|x|?Simplifying Radical Expressions: A principal square root cannot be negative.
In Simplifying Radical Expressions, what does product rule conditions mean?Simplifying Radical Expressions: A root-product rule under valid real assumptions.
Can √(a+b) usually split into √a+√b?Simplifying Radical Expressions: No, roots do not distribute across addition.
In Simplifying Radical Expressions, what does equivalent forms mean?Simplifying Radical Expressions: An alternate notation with the same value on a shared domain.
Why check domains after rewriting?Simplifying Radical Expressions: An algebraic rewrite may appear to accept an invalid input.
In Radical Operations, what does like radical addition mean?Radical Operations: Adding coefficients of matching radical terms.
Can √2 and √3 combine as 2√5?Radical Operations: No, their radicands differ.
In Radical Operations, what does multiplying radicals mean?Radical Operations: A product of radical expressions.
Why simplify after multiplication?Radical Operations: The new radicand may contain extractable perfect powers.
In Radical Operations, what does division and restrictions mean?Radical Operations: An equivalent quotient with domain safeguards.
Why state a denominator restriction?Radical Operations: Zero in any denominator makes the original expression undefined.
In Radical Functions and Inverses, what does square-root graph mean?Radical Functions and Inverses: A translated principal square-root function.
What is its least real input?Radical Functions and Inverses: The horizontal shift h.
In Radical Functions and Inverses, what does cube-root graph mean?Radical Functions and Inverses: The real odd-root family across every real x.
Why can negative inputs work?Radical Functions and Inverses: Odd powers of negative numbers are negative.
In Radical Functions and Inverses, what does inverse pair mean?Radical Functions and Inverses: Two functions with reversed allowed input-output pairs.
Why restrict x² to x≥0?Radical Functions and Inverses: Without it, different inputs give the same output.
In Radical Equations, what does isolate before squaring mean?Radical Equations: A staged method for a radical equation.
Why isolate the radical first?Radical Equations: Squaring a sum creates cross terms.
In Radical Equations, what does extraneous solutions mean?Radical Equations: A candidate introduced by a nonreversible step.
Where should candidates be checked?Radical Equations: In the unsquared original equation.
In Radical Equations, what does real-domain checks mean?Radical Equations: The allowable inputs and outputs of a real radical equation.
Can √x equal a negative real number?Radical Equations: No, its principal value is nonnegative.