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.

Licenses & Attributions