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
x^a * x^b = x^(a+b)
(x^a)^b = x^ab
x^(a/b) = the bth root of x^a, so
[x^(a/b)]^b = x^a
The laws work with all rational exponents
thank you
7x-5=72
Two laws of exponents are the Product Rule and the Quotient Rule.
1. Product Rule: When you multiply two terms with the same base, you can add their exponents.
Example: x^a * x^b = x^(a+b)
To simplify an expression using the Product Rule, you need to multiply the terms with the same base, and then add their exponents. For instance, let's consider the expression 3^2 * 3^3:
Step 1: Multiply the terms: 3^2 * 3^3 = 3^(2+3)
Step 2: Add the exponents: 3^(2+3) = 3^5
So, the simplified expression is 3^5.
2. Quotient Rule: When you divide two terms with the same base, you can subtract their exponents.
Example: x^a / x^b = x^(a-b)
To simplify an expression using the Quotient Rule, you need to divide the terms with the same base and then subtract their exponents. Let's take the expression (4^6 / 4^2):
Step 1: Divide the terms: 4^6 / 4^2 = 4^(6-2)
Step 2: Subtract the exponents: 4^(6-2) = 4^4
So, the simplified expression is 4^4.
Now, let's discuss how the laws of exponents work with rational (fractional) exponents. Rational exponents are equivalent to taking roots. If we have a fractional exponent, for example, x^(a/b), it is the same as the bth root of x raised to the power of a.
To simplify an expression with rational exponents, you need to apply the laws of exponents as usual but using the rules for roots. As an example, consider the expression (64^(2/3)):
Step 1: Apply the rational exponent: (64^(2/3)) = (cube root of 64)^2
Step 2: Calculate the cube root of 64: (cube root of 64) = 4
Step 3: Square the result: 4^2 = 16
So, the simplified expression is 16.
Now, let's provide a third expression that involves rational exponents: (27^(4/5)).
Step 1: Apply the rational exponent: (27^(4/5)) = (fifth root of 27)^4
Step 2: Calculate the fifth root of 27: (fifth root of 27) = 3
Step 3: Raise the result to the power of 4: 3^4 = 81
So, the simplified expression is 81.