The period (in seconds) of a pendulum with a length of
*
L
*
(in meters) is given by the formula
. If the length of a pendulum is
, what is its period?

### Guidance

When simplifying expressions with rational exponents, all the laws of exponents that were learned in the
*
Polynomial Functions
*
chapter are still valid. On top of that, the rules of fractions apply as well.

#### Example A

Simplify .

**
Solution:
**
Recall from the Product Property of Exponents, that when two numbers with the same base are multiplied we
*
add
*
the exponents. Here, the exponents do not have the same base, so we need to find a common denominator and then add the numerators.

This rational exponent does not reduce, so we are done.

#### Example B

Simplify

**
Solution:
**
This problem utilizes the Quotient Property of Exponents. Subtract the exponents with the same base and reduce
.

If you are writing your answer in terms of positive exponents, your answer would be . Notice, that when a rational exponent is improper we do not change it to a mixed number.

If we were to write the answer using roots, then we would take out the whole numbers. For example, can be written as because 6 goes into 19, 3 times with a remainder of 1.

#### Example C

Simplify .

**
Solution:
**
On the numerator, the entire expression is raised to the
power. Distribute this power to everything inside the parenthesis. Then, use the Powers Property of Exponents and rewrite 4 as
.

Combine like terms by subtracting the exponents.

Finally, rewrite the answer with positive exponents by moving the 2 and into the denominator.

**
Intro Problem Revisit
**
Substitute
for
*
L
*
and solve.

Therefore, the period of the pendulum is .

### Guided Practice

Simplify each expression. Reduce all rational exponents and write final answers using positive exponents.

1.

2.

3.

#### Answers

1. Change 4 and 8 so that they are powers of 2 and then add exponents with the same base.

2. Subtract the exponents. Change the power to .

3. Distribute the power to everything inside the parenthesis and reduce.

### Explore More

Simplify each expression. Reduce all rational exponents and write final answer using positive exponents.

- Rewrite your answer from Problem #1 using radicals.
- Rewrite your answer from Problem #4 using radicals.
- Rewrite your answer from Problem #4 using one radical.