**Roots and Rational Exponents**

Recall that can be defined as follows:

1. If is odd, then is the number such that .

2. If is even and , then is the number such that . (Notice that is undefined if is even and .

We can define rational exponents in the following manner:

If is a fraction reduced to lowest terms, then

,

assuming that if is even.

**Ex 1** Solve the equation
.

**Sol** Squaring both sides gives
, and then
subtracting from both sides gives . Then factoring gives
, so or . However, does not check in the
original equation, so is the only solution.

**Ex 2** Solve the equation
.

**Sol a** Adding 6 to both sides gives
, and then
factoring gives
, so either or
and therefore or .

**Sol b** Adding to both sides gives , and
then squaring both sides gives ; so
or .

**Pr 1** Simplify the expression
.

**Pr 2** Solve the equation .

**Pr 3** Solve the equation .

**Pr 4** Solve the following equations:

a) . b) .

**Pr 5** Find all values of for which the equation
is valid.

**Pr 6** Rewrite the expression
as a sum of
terms with rational exponents.

**Pr 7** Solve the equation
.

**Pr 8** Solve the equation
.

**Pr 9** Solve the equation
.

**Pr 10** Solve the equation
.

**Pr 11** Solve the equation
.

Go to the Solutions.

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