**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.

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