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Other Roots and Radicals / Domains / Notes (page 2 of 4)

Sections: Square roots, Other roots / Domains, Further simplifying, Rationalizing denominators


So far we have dealt only with square roots, but you would manipulate other radicals in much the same way. However, we need first to cover the notation for these other radicals.

Non-square-root radicals need a number to indicate which root is meant. By default, the simple radical symbol, "radical symbol", is assumed to mean "the square root". This is similar to exponents: if you have "x2", you know this means "x squared"; if you have just plain old "x", you know that the exponent is "1". Whenever there is no exponent, the exponent is understood to be "1"; to indicate other powers, you insert a number. In the same way, when there is no number on the radical, the radical is understood to be the square root; to indicate other roots, you insert a number, called the "index", inside that little "hook" on the front of the radical symbol. For instance:

    a square (second) root is written as radical symbol

    a cube (third) root is written as cbrt()

    a fourth root is written as  fourth-root()

    a fifth root is written as:  fifth-root()

...and so on.   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

When writing a square root, you can put a teeny "2" in, if you like, but this would be considered non-standard notation.

The process of simplification in these other roots works similarly to simplification of square roots. If you have a cube root, you can take out any factor that occurs in threes:

  • 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)

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.


Usually, we cannot have a negative inside a square root. (The exception is for "imaginary" numbers. If you haven't done the number "i" yet, then you haven't done imaginaries.) So, for instance, sqrt(-4) is not possible. Do not try to say something like " sqrt(-4) = -2 ", because it's not true: (-2)^2 = +4, which does not equal -4 . You must have a positive inside the square root. For instance:

  • Find the domain of the following:
    • y = sqrt(x - 2)

    The fact that I have the expression x – 2 inside a square root requires that x – 2 be zero or greater, so I must have x – 2 > 0. Solving, I get:

      domain:  x > 2

On the other hand, you CAN have a negative inside a cube root (or any other odd root). For instance:

    cbrt(-8) = -2

...because (–2)3 = –8.

  • Find the domain of the following:
    • y = cbrt(x - 2)

    For cbrt(x - 2), there is NO RESTRICTION on the value of x, because x – 2 is welcome to be negative inside a cube root. Then the domain is:

      domain:  all x


Do not confuse "simplifying" with "solving". If you have "x2 = 4", and you take the square root of either side, you will get x = ± 2, because you could square either of – 2 and +2 to get +4. But sqrt(4) does not equal ±2 , because "sqrt(4)" is specifically the positive square root. That is:

    sqrt(4) = 2

...and:

    -sqrt(4) = -2

Warning: Do not confuse these. Finding all the given roots of an equation (such as finding all the solution to the equation "x2 = 4") is different from simplifying an expression (such as "sqrt(4)"), because the former uses all the roots but the latter uses only the "principal" root.

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Cite this article as:

Stapel, Elizabeth. "Other Roots and Radicals / Domains / Notes." Purplemath. Available from
    http://www.purplemath.com/modules/radicals2.htm. Accessed
 

 

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