Properties of Radicals

A radical is the opposite of an exponent. As an example, if a number is squared, the exponent is 2. If the square root of a number is taken, the number is placed under a radical sign. The radical notation, "n(radical sign)x" represents the solution of the equation (x^n) where n is the exponent of the variable x. If the variable x in this case is negative then, the radical is undefined. If it is positive, then the radical solution is also positive. Properties of radicals can be used to solve algebraic problems involving radical expressions.
  1. Division Property

    • The division property of radicals can be used for different types of square root division. Square roots can be divided using the following property: sqrt(a/b) = sqrt(a)/sqrt(b) where a and b are positive real numbers. As an example, sqrt(1/16) can be simplified to sqrt(1)/sqrt(16) which is equal to 1/4.

    Simple Radical Form

    • There are three properties of simple radical form. Perfect squares should be factored out of a radical expression, fractions should not be left under a radical and the denominator of a fraction should not contain a radical. As an example, 1/(sqrt(3)) is not a simple radical because it contains a radical in the denominator. To reduce 1/((sqrt(3)) to its simple radical form, multiply the numerator and denominator with sqrt(3). This gives you sqrt(3)/((sqrt(3) * sqrt(3)) = sqrt(3)/3.

      Sqrt(3)/3 is a simple radical. It does not contain a perfect square, have a fraction under a radical or contain a radical in the denominator.

    Multiplication Property

    • Radical multiplication can be simplified with the use of the multiplication property. this property states that the square root of a variable multiplied with the square root of another variable is equal to the square root of the two variables multiplied together. Using the variables "a" and "b" it is represented as follows: sqrt(a) * sqrt(b) = sqrt( a * b). As an example, the equation, "sqrt(5)*sqrt(3)" is equal to "sqrt(15)."

    Fractional Property

    • Fractional exponents can be represented with radicals using the following property: x^(a/b) = (b(radical(x))^a. As an example, 5^(3/2) is equal to (sqrt(5))^3. This property can be used to simplify arithmetic equations. For example, "x * y^(1/3)" can be simplified to "x*3radical(y)."

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