Add, Subtract, Multiply, Divide Radical Expressions

Definition:  A radical expression is defined as any expression containing a radical (√) symbol. Many people mistakenly call this a ‘square root’ symbol, and many times it is used to determine the square root of a number. However, it can also be used to describe a cube root, a fourth root or higher.

Addition and Subtraction
To add or subtract radical expressions, combine any like radicals (terms with matching radicands and indexes). If there are no like radicals, expressions may be manipulated to help us combine them. Radical expressions can only be added or subtracted if they have like radicals.

Example of like radicals: 3√2x, √2x

Non-example of like radicals: ⁴√7a, ⁴√7b

To multiply radical expressions, use the product property of radicals, distributive property, or FOIL method and simplify the radicands.

Product property of radicals: ⁿ√a • ⁿ√b = ⁿ√ab

To divide radical expressions, use the quotient rule for radicals and rationalize the denominator to simplify if necessary.

Quotient rule for radicals:

Rationalizing the denominator:

If an expression contains a radical in the denominator, it is not simplified. To do so, multiply by a fraction equal to 1 that will eliminate the radical in the denominator. When the radical in the denominator is part of a binomial, rationalizing the denominator involves multiplying by a fraction equal to 1 made up of the conjugate.

Example: Add the following:

png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Cfrac%7B%5Csqrt%5B4%5D%7B16x%7D%7D%7B5%7D + %5Cfrac%7B3%5Csqrt%5B4%5D%7Bx%7D%7D%7B5%7D


Step 1: Simplify radicals

Taking the fourth root of 16 in the first term allows us to rewrite this as:
png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Cfrac%7B%5Csqrt%5B4%5D%7B16x%7D%7D%7B5%7D + %5Cfrac%7B3%5Csqrt%5B4%5D%7Bx%7D%7D%7B5%7D%5C%5C%5C%5C%3D%5Cfrac%7B%5Csqrt%5B4%5D%7B16%7D%5Csqrt%5B4%5D%7Bx%7D%7D%7B5%7D + %5Cfrac%7B3%5Csqrt%5B4%5D%7Bx%7D%

Step 2: Combine like radicals

Because these radicals have matching indexes and radicands, we can rewrite this as:
png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Cfrac%7B2%5Csqrt%5B4%5D%7Bx%7D%7D%7B5%7D + %5Cfrac%7B3%5Csqrt%5B4%5D%7Bx%7D%7D%7B5%7D%5C%5C%5C%5C%3D %5Cfrac%7B2%5Csqrt%5B4%5D%7Bx%7D+3%5Csqrt%5B4%5D%7Bx%7D%7D%7B5%7D%5C%5C%5C%5C%3D%5Cfrac%7B5%5Csqrt%

Step 3: Simplifypng.latex-%5Cdpi%7B150%7D %5Cbg_white %5Cfrac%7B5%5Csqrt%5B4%5D%7Bx%7D%7D%7B5%7D%5C%5C%5C%5C%3D%5Csqrt%5B4%5D%7Bx%7D

Radicals – where a variable is inside a square root, cube root, or otherwise – can be tricky! Watch and see how a MathCrunch tutor helps a student like you solve a radical equation.

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