Equations Involving Rational Exponents
Example
An equation with no solution
Solve (2t - 3)-2/3 = -1.
Solution
Raise each side to the power -3 to eliminate the root and the negative sign in the
exponent:
| (2t - 3)-2/3 |
= -1 |
Original equation |
| [(2t - 3)-2/3]-3 =
(-1)-3 |
= (-1)-3 |
Raise each side to the -3 power. |
| (2x - 3)2 |
= -1 |
Multiply the exponents:
 |
By the even-root property this equation has no real solution. The square of every
real number is nonnegative.
Summary of Methods
The three most important rules for solving equations with exponents and radicals
are restated here.
Strategy for Solving Equations with
Exponents and Radicals
1. In raising each side of an equation to an even power, we can create an equation
that gives extraneous solutions. We must check all possible solutions in
the original equation.
2. When applying the even-root property, remember that there is a positive and
a negative even root for any positive real number.
3. For equations with rational exponents, raise each side to a positive or
negative integral power first, then apply the even- or odd-root property.
(Positive fraction—raise to a positive power; negative fraction—raise to a
negative power.)
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