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?
give examples to indicate these laws:
A) (AB)C= ABC
b. A^B * A^D= AB+D
Two laws of exponents are the product rule and the power rule. These laws are used to simplify expressions involving exponents.
1. Product Rule: When multiplying two exponential expressions with the same base, you can add the exponents.
Example: If you have x^3 * x^4, you can simplify it as x^(3+4) = x^7.
To simplify an expression using the product rule, you identify terms with the same base and then add their exponents together.
2. Power Rule: When raising an exponential expression to another exponent, you can multiply the exponents.
Example: If you have (x^2)^3, you can simplify it as x^(2*3) = x^6.
To simplify an expression using the power rule, you raise the base to the product of the exponents.
When it comes to rational exponents, the laws of exponents still hold true. For instance, if you have x^(3/2) * x^(2/3), you can apply the product rule to simplify it as x^((3/2) + (2/3)). To perform the addition with rational exponents, you need a common denominator; in this case, it would be 6. So, the expression becomes x^((9/6) + (4/6)) = x^(13/6).
Similarly, you can use the power rule with rational exponents. For example, (x^(1/2))^3 simplifies to x^((1/2)*3) = x^(3/2).
In both cases, the laws of exponents extend to rational exponents by applying the principles of adding and multiplying fractions.