Summary: Radicals and Rational Exponents

Key Concepts

  • The principal square root of a number [latex]a[/latex] is the nonnegative number that when multiplied by itself equals [latex]a[/latex].
  • If [latex]a[/latex] and [latex]b[/latex] are nonnegative, the square root of the product [latex]ab[/latex] is equal to the product of the square roots of [latex]a[/latex] and [latex]b[/latex]
  • If [latex]a[/latex] and [latex]b[/latex] are nonnegative, the square root of the quotient [latex]\frac{a}{b}[/latex] is equal to the quotient of the square roots of [latex]a[/latex] and [latex]b[/latex]
  • We can add and subtract radical expressions if they have the same radicand and the same index.
  • Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.
  • The principal nth root of [latex]a[/latex] is the number with the same sign as [latex]a[/latex] that when raised to the nth power equals [latex]a[/latex]. These roots have the same properties as square roots.
  • Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals.
  • The properties of exponents apply to rational exponents.

Glossary

index the number above the radical sign indicating the nth root

principal nth root the number with the same sign as [latex]a[/latex] that when raised to the nth power equals [latex]a[/latex]

principal square root the nonnegative square root of a number [latex]a[/latex] that, when multiplied by itself, equals [latex]a[/latex]

radical the symbol used to indicate a root

radical expression an expression containing a radical symbol

radicand the number under the radical symbol

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