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

Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains


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

Simplifying Higher-Index Terms

  • Simplify fourth root of 16
  • 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:

      4th-rt[16] = 4th-rt[2*2*2*2] = 2

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:  cbrt(8)
    • cbrt(8) = cbrt(2 * 2 * 2) = 2

  • Simplify the cube root:  cbrt(54)
    • cbrt(54) = cbrt(2 * 3 * 3 * 3) = cbrt(3 * 3 * 3) cbrt(2) = 3 cbrt(2)

  • Simplify:  cbrt[48]   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved
    • cbrt[48] = cbrt[3*2*2*2*2] = 2 cbrt[3*2] = 2 cbrt[6]

  • Simplify:  4 cbrt[27]
    • 4 cbrt[27] = 4 cbrt[3*3*3] = 4 * 3 = 12

  • Simplify:  5th-rt[32 x^10 y^6 z^7]
    • 5th-rt[32 x^10 y^6 z^7] = 5th-rt[2*2*2*2*2 * x^2 x^2 x^2 x^2 x^2 * yyyyy * y * zzzzz * z^2] = 2 x^2 y z * 5th-rt[yz^2]


Multiplying Higher-Index Roots

  • Simplify:  cbrt[9] cbrt[24]
    • cbrt[9] cbrt[24] = cbrt[3 * 3 * 3 * 2 * 2 * 2] = 3 * 2 = 6

  • Simplify:  4th-rt[75] (2 * 4th-rt[100])
    • 4th-rt[75] (2 * 4th-rt[100]) = 2 * 4th-rt[3 * 5 * 5 * 2 * 2 * 5 * 5] = 2 * 5 * 4th-rt[2 * 2 * 3] = 10 * 4th-rt[12]


Adding Higher-Index Roots

  • Simplify:  cbrt[8] + cbrt[64]
    • cbrt[8] + cbrt[64] = cbrt[2 * 2 * 2] + cbrt[4 * 4 * 4] = 2 + 4 = 6

  • Simplify:  cbrt[81] + 5*cbrt[3]
    • cbrt[81] + 5*cbrt[3] = cbrt[3*3*3 * 3] + 5*cbrt[3] = 3*cbrt[3] + 5*cbrt[3] = 8 cbrt[3]


Dividing Higher-Index Roots

  • Simplify:  cbrt[ 5 / 27 ]
    • cbrt[5/27] = cbrt[5] / cbrt[27] = cbrt[5] / cbrt[3*3*3] = cbrt[5] / 3

  • Simplify:  cbrt[ 27 / 5 ]
  • I can't simplify this expression properly, because I can't simplify the radical in the denominator down to whole numbers:

      cbrt[ 27 / 5 ] = cbrt[3*3*3] / cbrt[5] = 3 / cbrt[5]

    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:

      (3 / cbrt[5])*(cbrt[5*5] / cbrt[5*5]) = (3 cbrt[25]) / cbrt[5*5*5] = (3 cbrt[25]) / 5

  • Simplify:  4th-rt[ 5 / 72 ]
  • 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:

      (4th-rt[5] / 4th-rt[2*2*2*3*3])*(4th-rt[2*3*3] / 4th-rt[2*3*3]) = 4th-rt[5*2*3*3] / 4th-rt[2*2*2*2*3*3*3*3] = 4th-rt[90] / (2 * 3) = 4th-rt[90] / 6

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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
 

 

 

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