Higher-Index Roots: Basic Operations (page 6 of 7)

Operations with cube roots, fourth roots, and other higher-index roots work similarly to square roots.

Simplifying Higher-Index Terms

• Simplify
• Just as I can pull from a square (or second) root anything that I have two copies of, so also I can pull from a fourth root anything I've got four of:

If you have a cube root, you can take out any factor that occurs in threes; in a fourth root, take out any factor that occurs in fours; in a fifth root, take out any factor that occurs in fives; etc.

• Simplify the cube root:
• Simplify the cube root:
• Simplify:
• Simplify:

Multiplying Higher-Index Roots

• Simplify:
• Simplify:

• Simplify:
• Simplify:

Dividing Higher-Index Roots

• Simplify:
• Simplify:
• I can't simplify this expression properly, because I can't simplify the radical in the denominator down to whole numbers:

To rationalize a denominator containing a square root, I needed two copies of whatever factors were inside the radical. For a cube root, I'll need three copies. So that's what I'll multiply onto this fraction:

• Simplify:
• Since 72 = 8 × 9 = 2 × 2 × 2 × 3 ×3, I won't have enough of any of the denominator's factors to get rid of the radical. To simplify a fourth root, I would need four copies of each factor. For this denominator's radical, I'll need two more 3s and one more 2:

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 Cite this article as: Stapel, Elizabeth. "Higher-Index Roots: Basic Operations." Purplemath. Available from     http://www.purplemath.com/modules/radicals6.htm. Accessed [Date] [Month] 2016

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