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.