Lesson 3: 7-3 Rational Exponents
Duration of Days: 3
Lesson Objective
Evaluate and rewrite expressions involving rational exponents.
Solve equations involving expressions with rational exponents.
How can the exponent property (x^m)^n = x^{m • n} be used to show that x^{1/2} must be equivalent to v{x}? (Hint: Consider what happens when you raise x^{1/2} to the power of 2.) What does the denominator of a rational exponent always represent in a radical expression?
In a rational exponent like a^{p/q}, what does the numerator (p) tell you to do to the base, and what does the denominator (q) tell you to do? Can you perform the root first, or the power first, and why does the result remain the same? (e.g., compare (8^{1/3})^2 to (8^2)^{1/3})
When multiplying expressions with rational exponents (e.g., x^{1/3} • x^{1/6}), you must still add the exponents. How is the process of adding rational exponents similar to, and different from, adding whole number exponents, and what must you first do to the fractions to correctly apply the Product of Powers Property?
Rational exponent
Cube root
nth root
Exponential equation
N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
See below
Exponents can be fractions. For any real numbers a and b, and any positive integer n, if a^n=b, then a is an nth root of b. If a^n=b, then nvb=a. For any positive real number b, and any integers m and n with n>1, b^{m/n}=nv{b^m}.
The population p of a culture that begins with 40 bacteria and doubles every 8 hours can be modeled by p=40(2)^{t/8}, where t is the time in hours. Find t if p=20,480.
In exercise 92, students should recognize that the bases of the expressions must have equal bases to apply the power property of equality.
Interpersonal learners. Divide the class into groups of two or three students. Have students discuss what they knew about exponents before starting the lesson and how it relates to rational exponents. for example, if an exponent tells you how many times to use a base as a factor, what does an exponent of 3/2 represent? Encourage students to explore the concepts and ask each other for information.
Practice: Exercises 1 -16
Exercises 94-102