What are Radical Expressions?

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  • Written By: Michael Smathers
  • Edited By: Jessica Seminara
  • Last Modified Date: 27 August 2019
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A radical expression in algebra is an expression that includes a radical, or root. These are the inverse operations to exponents, or powers. Radical expressions include added roots, multiplied roots and expressions with variables as well as constants. These expressions have three components: the index, the radicand, and the radical. The index is the degree taken, the radicand is the root being derived, and the radical is the symbol itself.

By default, a radical sign symbolizes a square root, but by including different indexes over the radical, cube roots, fourth roots or any whole number root can be taken. Radical expressions can include either numbers or variables under the radical, but the fundamental rules remain the same regardless. To work with radicals, the expressions must be in simplest form; this is accomplished by removing factors from the radicand.

The first step in simplifying radicals is breaking the radicand into the factors needed to equal the number. Then, any perfect square factors must be placed to the left of the radical. For example, √45 can be expressed as √9*5, or 3√5.

To add radical expressions, the index and radicand must be the same. After these two requirements have been met, the numbers outside the radical can be added or subtracted. If the radicals cannot be simplified, the expression has to remain in unlike form. For example, √2+√5 cannot be simplified because there are no factors to separate. Both terms are in their simplest form.


Multiplying and dividing radical expressions works using the same rules. Products and quotients of radical expressions with like indexes and radicands can be expressed under a single radical. The distributive property works in the same fashion as it does with integer expressions: a(b+c)=ab+ac. The number outside the parenthesis should be multiplied by each term inside parenthesis in turn, retaining addition and subtraction operations. After all terms inside the distributive parentheses are multiplied, the radicals have to be simplified as usual.

Radical expressions that are part of an equation are solved by eliminating the radicals according to the index. Normal radicals are eliminated by squaring; therefore, both sides of the equation are squared. For example, the equation √x=15 is solved by squaring the square root of x on one side of the equation and 15 on the right, yielding a result of 225.


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