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?
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Two laws of exponents are the product law and the power law.
1. Product Law: When you multiply two exponential expressions with the same base, you can add their exponents.
Example: 3^4 * 3^2 = 3^(4+2) = 3^6.
To simplify expressions using the product law, identify the common base and add the exponents.
2. Power Law: When you raise an exponential expression to another exponent, you can multiply the exponents.
Example: (2^3)^2 = 2^(3*2) = 2^6.
To simplify expressions using the power law, apply the exponent outside the parentheses to each term inside.
Now, let's talk about rational exponents. A rational exponent is a fractional exponent, representing the root or the reciprocal of the base raised to a certain power. The laws of exponents apply to rational exponents in the same way.
For instance, consider the expression (4^(1/2))^2. Using the power law, we multiply the exponents: 4^(1/2 * 2) = 4^1 = 4.
Similarly, if we have the expression (8^(1/3))^4, we apply the power law: 8^(1/3 * 4) = 8^(4/3).
To simplify further, we can write 8^(4/3) as (∛8)^4, since the cube root of 8 is 2. Therefore, (∛8)^4 = 2^4 = 16.
In summary, the laws of exponents, such as the product law and power law, allow us to simplify expressions involving exponents. They also extend to rational exponents, making it possible to manipulate and simplify expressions involving fractional powers.