# Rational Exponents: Extending Properties of Integer Exponents

Rational Exponents: Extending Properties of Integer Exponents
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Slide 1: Slide

This lesson contains 19 slides, with interactive quizzes and text slides.

## Items in this lesson

Rational Exponents: Extending Properties of Integer Exponents

#### Slide 1 -Slide

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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.

#### Slide 2 -Slide

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

#### Slide 3 -Mind map

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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.

#### Slide 4 -Slide

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.

#### Slide 5 -Slide

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

#### Slide 6 -Slide

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).

#### Slide 7 -Slide

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.

#### Slide 8 -Slide

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.

#### Slide 9 -Slide

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)

#### Slide 10 -Slide

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))

#### Slide 11 -Slide

Give students time to practice simplifying expressions with rational exponents and provide feedback.
We can also use rational exponents as a notation for radicals. For example, √4 can be written as 4^(1/2).

#### Slide 12 -Slide

Explain how rational exponents can be used as a 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)

#### Slide 13 -Slide

Provide examples of how to use rational exponents as a notation for radicals.
Write the following radicals using rational exponents:
1. ∛64
2. ⁵√243
3. ∜81

#### Slide 14 -Slide

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.

#### Slide 15 -Slide

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.

#### Slide 16 -Slide

Summarize the key concepts of the lesson and emphasize their importance.
Write down 3 things you learned in this lesson.

#### Slide 17 -Open question

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.

#### Slide 18 -Open question

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