Rational Exponents: Extending Properties of Integer Exponents

Rational Exponents: Extending Properties of Integer Exponents

Learning Objective

At the end of the lesson you will be able to 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.

Introduce the learning objective and explain what the students will be able to do by the end of the lesson.

What do you already know about integer exponents and radicals?

Recap: Integer Exponents

Integer exponents are a shorthand notation for multiplying a number by itself a certain number of times. For example, 2^3 means 2 multiplied by itself three times: 2 x 2 x 2 = 8.

Remind students of what integer exponents are and how they are used.

Properties of Integer Exponents

There are several properties of integer exponents, including the product rule, quotient rule, and power rule. These properties allow us to simplify expressions with exponents.

Explain the importance of the properties of integer exponents and how they are used.

Radicals

A radical is another way of representing the root of a number. For example, the square root of 4 can be written as √4 or as 4^(1/2).

Introduce the concept of radicals and how they are used.

Rational Exponents

Rational exponents are another way of expressing radicals. For example, the cube root of 8 can be written as 8^(1/3).

Explain what rational exponents are and how they relate to radicals.

Extending Properties

We can extend the properties of integer exponents to rational exponents by using the same rules. For example, (a^m)^n = a^(mn) can be extended to (a^(m/n))^n = a^m.

Show how the properties of integer exponents can be extended to rational exponents.

More Properties

Other properties of rational exponents include the product rule, quotient rule, and power rule. These rules allow us to simplify expressions with rational exponents.

Explain the additional properties of rational exponents and how they are used.

Examples: Simplifying Expressions

Here are some examples of simplifying expressions with rational exponents:
1. (x^(1/3))^2 = x^(2/3)
2. (2^(1/2) * 3^(1/2))^2 = 6
3. (a^(2/3) * a^(1/3)) / a = a^(2/3)

Provide examples of how to simplify expressions with rational exponents.

Practice: Simplifying Expressions

Simplify the following expressions using rational exponents:
1. (y^(1/2))^3
2. (4^(1/3) * 2^(1/3))^3
3. (b^(1/4))^2 / (b^(1/2))

Give students time to practice simplifying expressions with rational exponents and provide feedback.

Notation for Radicals

We can also use rational exponents as a notation for radicals. For example, √4 can be written as 4^(1/2).

Explain how rational exponents can be used as a notation for radicals.

Examples: Notation for Radicals

Here are some examples of using rational exponents as a notation for radicals:
1. ∛27 = 27^(1/3)
2. ⁴√625 = 625^(1/4)
3. ∜16 = 16^(1/4)

Provide examples of how to use rational exponents as a notation for radicals.

Practice: Using Notation for Radicals

Write the following radicals using rational exponents:
1. ∛64
2. ⁵√243
3. ∜81

Give students time to practice using rational exponents as a notation for radicals and provide feedback.

Real-World Applications

Rational exponents and radicals are used in many real-world applications, including engineering, physics, and finance.

Discuss some real-world applications of rational exponents and how they are used.

Conclusion

Rational exponents allow us to extend the properties of integer exponents to include fractional values, providing a notation for radicals. By understanding these concepts and their applications, we can solve complex problems in various fields.

Summarize the key concepts of the lesson and emphasize their importance.

Write down 3 things you learned in this lesson.

Have students enter three things they learned in this lesson. With this they can indicate their own learning efficiency of this lesson.

Write down 2 things you want to know more about.

Here, students enter two things they would like to know more about. This not only increases involvement, but also gives them more ownership.

Ask 1 question about something you haven't quite understood yet.

The students indicate here (in question form) with which part of the material they still have difficulty. For the teacher, this not only provides insight into the extent to which the students understand/master the material, but also a good starting point for the next lesson.