Skip to main content
Back

Multiplying, Dividing, and Rationalizing Radicals quiz

Control buttons has been changed to "navigation" mode.
1/15
  • What does it mean to rationalize the denominator?
    It means to eliminate any radicals from the denominator of a fraction so that the denominator is a rational number.
  • Why can't you leave a radical in the denominator of a fraction?
    Because in standard mathematical form, denominators should be rational, not irrational or containing radicals.
  • What do you multiply by to rationalize a denominator with a single radical term, like 1/√3?
    You multiply both the numerator and denominator by the same radical, in this case √3.
  • What is the result of multiplying √3 by √3?
    The result is 3, because √3 × √3 = √9 = 3.
  • After rationalizing 1/√3, what is the equivalent expression?
    The equivalent expression is √3/3.
  • What is a conjugate in the naturally occurring context of rationalizing denominators?
    A conjugate is formed by changing the sign between two terms in a binomial, such as turning 2 + √3 into 2 - √3.
  • Why do you use the conjugate to rationalize denominators with two terms, like 1/(2+√3)?
    Because multiplying by the conjugate creates a difference of squares, which eliminates the radical in the denominator.
  • What is the general formula for the conjugate of a + √b?
    The conjugate is a - √b.
  • What happens when you multiply (2+√3) by its conjugate (2-√3)?
    You get a difference of squares: (2)^2 - (√3)^2 = 4 - 3 = 1.
  • When rationalizing 1/(2+√3), what do you multiply the numerator and denominator by?
    You multiply both by the conjugate, which is (2-√3).
  • What is the simplified result of rationalizing 1/(2+√3)?
    The result is (2-√3)/1, which simplifies to 2-√3.
  • Why does multiplying by the conjugate always eliminate the radical in the denominator?
    Because it uses the difference of squares, which cancels out the radical terms.
  • What must you always do to both the numerator and denominator when rationalizing?
    You must multiply both by the same value to keep the expression equivalent.
  • Is it acceptable to have a radical in the numerator after rationalizing?
    Yes, having a radical in the numerator is acceptable; only the denominator must be rational.
  • What are the two main cases for rationalizing denominators?
    If the denominator is a single term, multiply by the radical; if it's two terms, multiply by the conjugate.