Roots and Rational Exponents
Sol 1
Sol 2a gives , so factoring yields . Therefore either or , but since is never negative; so and therefore .
Sol 2b Squaring both sides of gives or , so and . Therefore or , but is the only solution since does not check in the original equation.
Sol 3a gives , so factoring gives . Then or , but since is never negative; so and therefore .
Sol 3b gives , so squaring both sides gives . Then or . The answer does not check in the original equation, though, so is the only answer.
Sol 4
a) Cubing both sides of gives , so taking the square root of both sides gives .
b) Squaring both sides of gives , so taking the cube root of both sides gives .
Sol 5 Squaring both sides of gives , so subtracting from both sides gives and so . Therefore this equation is valid only for .
Sol 6 .
Sol 7 Cubing both sides of gives , so or or .
Sol 8 or . However, does not check in the original equation, so is the only solution.
Sol 9 Squaring both sides of gives , so and therefore or . Squaring both sides of this equation yields , so and . Since this answer checks in the original equation, it is the only solution.
Sol 10 Squaring both sides of gives , so and therefore . Squaring both sides of this equation gives , so and therefore . Factoring gives , so or . However, does not check in the original equation, so is the only solution.
Sol 11 Raising both sides of the equation to the 6th power gives or .
Multiplying out both sides gives , so and therefore and .
Then , so either or . Completing the square in the last equation gives or , so and .
However, does not check in the original equation (since when and therefore while ). Therefore and are the only solutions.
Return to the Problems for this
Topic
Return to the Precalculus Home Page