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 .