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We are about to consider expressions involving variables inside of square roots. In order to make the simplification rules simpler, and to avoid a discussion of the "domain" of the square root, we assume that all variables represent non-negative real numbers.

Simplifying square roots with variables is similar to simplifying square roots without variables. Treat the variable as a factor--if it appears twice (x2), cross out both and write the factor (x) one time to the left of the square root sign. If the factor appears three times (x3), treat this as x2×x: cross out x2 and write x to the left of the square root sign, leaving the single x inside the square root sign.

In general, follow these rules:

  • If the exponent of the variable is even, divide the exponent by two and write the result to the left of the square root sign, leaving no variable inside the square root sign.
  • If the exponent of the variable is odd, subtract one from the exponent, divide it by two, and write the result to the left of the square root sign, leaving the variable inside the square root sign once, with no exponent.

Example 1: Simplify

  1. Factor: =
  2. Write factors outside sign: = 3×x
  3. Multiply numbers outside sign: 3×x = 3x
  4. Check: (3x)2(5) = (3)2(x)2(5) = 45x2.
Thus, = 3x. Notice our assumption that x is a non-negative real number is essential; both and 3x make sense for x < 0, but they are not equal in this case.

Example 2: Simplify

  1. Factor: =
  2. Write factors outside sign: = 2×2×x×y
  3. Multiply numbers outside sign: 2×2×x×y = 4xy
  4. Check: (4xy)2(5x) = (4)2(x)2(y)2(5x) = 80x3y2.
Thus, = 4xy.

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