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How to Subtract Radicals When Indexes Are Not the Same

Subtracting expressions containing radicals is a common exercise in high school and college algebra and calculus classrooms. A radical expression is one in which the "root" of a certain value is taken. When you subtract or add radical expressions, the bases must be the same. However, if the indexes (numbers on the upper left side of the radical sign) are different, you must approach the problem from an alternative angle. Although you cannot subtract radicals with different indexes when the bases are variables (letters that represent values), you can perform this task if your bases are numbers.

Instructions

- 1
  Write the subtraction expression. For example, you might have ^3√64 - ^4√16 (the cube root of 64 minus the fourth root of 16).
- 2
  Simplify each separate radical expression. In this case, you must find out what number multiplied times itself three times equals 64. The answer for the cube root of 64 is 4. You also must figure out what number multiplied times itself equals 16. The answer, therefore, for the fourth root of 16 is 2. You can determine roots of your base by making guesses and then performing the multiplications to check your answers, or you can use online radical/root calculator tools (see Resources).
- 3
  Rewrite the subtraction expression with the roots that you obtained in step 2. In this example, you would write 4 - 2.
- 4
  Subtract the second value from the first to find your answer. In this case, you would subtract 2 from 4 to get 2.

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