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 (x^{2}), cross out both and write the
factor (x) one time to the left of the square root sign. If the
factor appears three times (x^{3}), treat this as x^{2}×x:
cross out x^{2} 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
- Factor: =
- Write factors outside sign: = 3×x
- Multiply numbers outside sign: 3×x = 3x
- Check: (3x)^{2}(5) = (3)^{2}(x)^{2}(5) = 45x^{2}.
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
- Factor: =
- Write factors outside sign: = 2×2×x×y
- Multiply numbers outside sign: 2×2×x×y = 4xy
- Check: (4xy)^{2}(5x) = (4)^{2}(x)^{2}(y)^{2}(5x) = 80x^{3}y^{2}.
Thus,
= 4xy.