Describe two laws of exponents and provide an example illustrating each law. Explain how to simplify your expression. How do the laws work with rational exponents? Provide the class with a third expression to simplify that includes rational (fractional) exponents.

Describe two laws of exponents and provide an example illustrating each law. Explain how to simplify your expression. How do the laws work with rational exponents? Provide the class with a third expression to simplify that includes rational (fractional) exponents.

Two laws of exponents are the product rule and the power rule.

1. Product Rule: To multiply two exponential expressions with the same base, you add their exponents.
Example: If we have the expression x^a * x^b, where "x" is the base and "a" and "b" are the exponents, the product rule states that x^a * x^b = x^(a + b).

2. Power Rule: To raise an exponential expression to another exponent, you multiply the exponents together.
Example: If we have the expression (x^a)^b, where "x" is the base, "a" is the exponent of the base, and "b" is the exponent of the exponent, the power rule states that (x^a)^b = x^(a * b).

To simplify an expression using these rules, follow these steps:
1. Identify if the base is the same in all terms.
2. If the base is the same, use the product rule to add the exponents when multiplying, or the power rule to multiply exponents when raising to another exponent.
3. Combine like terms, if necessary.
4. Simplify any remaining expressions.

The laws of exponents work similarly with rational exponents. A rational exponent represents taking the nth root of a number. Here's an example:

3. Third Expression: Simplify the expression (x^(1/3))^2.
In this expression, the base (x) is raised to the exponent of 1/3, and then the result is raised to the exponent of 2.
According to the laws of exponents, we can apply the power rule: (x^(1/3))^2 = x^(1/3 * 2) = x^(2/3).
The simplified expression is x^(2/3), which means taking the cube root of x and then squaring the result.

Remember, these laws help simplify expressions, making them easier to work with and understand.