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.
Laws of exponents:
xa*xb = xa+b
xa/xb = xa-b
(xa)b = xab
x-a = 1/xa
x1/a = ath root of x
Some or all of these laws must have been discussed in class. Feel free to create the example and post here for discussions if you wish.
Two laws of exponents are the product law and the power law.
1. Product Law: When multiplying two expressions with the same base, you add the exponents.
For example, if you have x^2 * x^3, you can simplify it using the product law. You add the exponents: 2 + 3 = 5. So, x^2 * x^3 simplifies to x^5.
2. Power Law: When raising a power to another power, you multiply the exponents.
For example, if you have (x^2)^3, you can simplify it using the power law. You multiply the exponents: 2 * 3 = 6. So, (x^2)^3 simplifies to x^6.
To simplify an expression with exponents, you'll want to follow these steps:
1. Identify if you have any like terms with the same base.
2. If you have like terms, apply the product or power law to simplify them.
3. Combine any remaining terms by adding or subtracting them.
The laws of exponents are also applicable to expressions with rational (fractional) exponents.
For instance, let's consider the expression y^(3/2) * y^(1/2). To simplify this expression, we add the exponents: (3/2) + (1/2) = (4/2) = 2. So, y^(3/2) * y^(1/2) simplifies to y^2.
Additionally, let's take a look at the expression (z^(1/3))^2. To simplify this expression, we multiply the exponents: (1/3) * 2 = 2/3. So, (z^(1/3))^2 simplifies to z^(2/3).
In summary, the laws of exponents allow us to simplify expressions by either adding or multiplying exponents when dealing with like terms. These laws also extend to rational exponents, where the same rules apply.