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Simplifying Radical Expressions quiz

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  • What is the first step to simplify a radical like √20?
    Break the radicand into a product of factors, aiming for one to be a perfect square.
  • How do you simplify √20?
    √20 = √4 × √5 = 2√5, since 4 is a perfect square and 5 is prime.
  • What property allows you to split a radical into two separate radicals?
    If a radicand has factors a and b, √(ab) = √a × √b.
  • How do you simplify √18x²?
    √18x² = √9 × √2 × √x² = 3x√2.
  • What is the cube root of 54x⁴ simplified?
    ∛54x⁴ = ∛27 × ∛2 × ∛x³ × ∛x = 3x∛2x.
  • How do you simplify radicals with variables, like √x³?
    Split the exponent: √x³ = √x² × √x = x√x.
  • What shortcut helps simplify radicals with variable exponents?
    Divide the exponent by the index; the quotient is the exponent outside, and the remainder stays under the radical.
  • How do you simplify √x⁷?
    √x⁷ = x³√x, since 2 goes into 7 three times with one left over.
  • How do you simplify √8x⁵?
    √8x⁵ = 2x²√2x, by factoring 8 as 4×2 and x⁵ as x⁴×x.
  • How do you simplify a radical with a fraction, like √(49/64)?
    Split into √49/√64 = 7/8, since both are perfect squares.
  • What is the result of combining √32 and √2?
    √32/√2 = √(32/2) = √16 = 4.
  • How do you simplify √(64x⁴/9x²)?
    √(64x⁴/9x²) = (8x²)/(3x) = (8/3)x.
  • What is the rule for adding or subtracting radical expressions?
    You can only combine radicals with the same radicand and index, called like radicals.
  • How do you add 3√7 + 2√7 - ∛7?
    Combine like radicals: 3√7 + 2√7 = 5√7; ∛7 is not like and stays separate.
  • What should you do before adding or subtracting unlike radicals?
    Simplify each radical to see if they can become like radicals, then combine if possible.