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Simplifying Radicals Involving Variables

Examples

In the following example we start with a square root of a quotient.

 

Example 1

Radicals with variables

Simplify each expression. Assume all variables represent positive real numbers.

Solution

Quotient rule for radicals
  Rationalize the denominator.
   
Quotient rule for radicals
  Product rule for radicals
  Simplify.
  Rationalize the denominator.
   

Any variable with an exponent that is a multiple of 3 is a perfect cube. For example, a3, b6, c15, and w39 are perfect cubes. Each of these expressions is the cube of a variable with an integral exponent. For any values of the variables we can write

Note that when we find the cube root, the result has one-third of the original exponent.

If the exponent on a variable is a multiple of 4, we have a perfect fourth power; if the exponent is a multiple of 5, we have a perfect fifth power; and so on. In the next example we simplify radicals with an index higher than 2.

 

Example 2

Simplifying higher-index radicals with variables

Simplify. Assume the variables represent positive numbers.

Solution

a) Use the product rule to place the largest perfect cube factors under the first radical and the remaining factors under the second:

b) Place the largest perfect fourth power factors under the first radical and the remaining factors under the second:

c) Multiply by to rationalize the denominator:

 
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