# Rational Exponent Worksheets

We have worked with our core competency as far as exponents so far which states that an exponent is simply a multiple of the root value by how ever many times it is raised to a power. What if that power was fraction? How would you adjust your calculations based on that? When working with a fractional exponent, the denominator of that exponent serves as the degree of radical that it creates. The purpose of the numerator is the power of the term found inside that radical. In order to proficiently use this skill, you should be well versed on the concepts of roots and anatomy of them. Rational exponents lend themself well to many different uses of the quadratic equations. There are also many different financial applications of this type of math that allow us to make reliable forecast and thereby allowing business leaders to make sound decisions. Economists use this technique often to compute interest rates and create various types of financial products. It is often used to understand the concepts of depreciation and inflation. This can help you assess the stunted or diminishing progression of just about anything with a quantitative value. This series of lessons and worksheets will help you better understand the core fundamentals of working with and using rational exponents.

### Aligned Standard: 6.EE.A.1

- Simplifying Expressions With Rational Exponents Lesson- In our problem the root value (5) is the same. This tells us that the multiplication property of rational exponents applies.
- Guided Lesson - Reduce the following expressions to their least complex form.
- Guided Lesson Explanation - We will look at work operations within in this seemingly complex expression set.
- Practice Worksheet - Break everything down to the lowest possible form.
- Simplifying Worksheet - You will approach some of the most common forms of operations that are applied to this skill.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Each of these sheets gets progressively more difficult.

- Homework 1 - You will work with fractional exponents and strive to make sense of them in a simple form.
- Homework 2 - You will advance to the next level and work exponent of fractional bases. It is not as difficult as it first appears.
- Homework 3 - We now add an operation that you must evaluate as well.
- Homework 4 - This can be complicated if do not plan out the steps that you will take along the way.
- Homework 5 - You will need to draw up the steps of your approach for this one too.

### Practice Worksheets

This selection of sheets will help students better understand the sequence they should use to solve these problems.

- Practice 1 - Write each expression as a radical in simplest form.
- Practice 2 - You may leave improper fractions in the exponent.
- Practice 3 - The focus here is end off with positive exponents.
- Practice 4 - Do not forget that an operation is found within every expression.

### Quiz Sheets

It is time to see how well you can manipulate this skill.

- Quiz 1 - You will rewrite these expressions in a simple form.
- Quiz 2 - We see how you handle negative rational exponents.
- Quiz 3 - The example is really good to help elevate your mindset for all the problems on the page.
- Quiz 4 - You may find that students try to solve for the unknown variables.

### How to Simplify Expressions with Rational Expressions

When we are working with calculations that involve rational exponents, we just need to remember our basic math operations with them. It all begins by understanding the vocabulary for these types of expressions. The best way to explain this is just to jump in and take a look at an expression. Say for instance we were evaluating the expression: 3x^{ ½}. In this instance 3 serves as the coefficient. The variable (x) is the base, and the rational exponent is the fraction (½). Now that we understand all the mechanics of these expressions and what we call their parts, lets evaluate how they work with operations. When we fully understand how these operations work, we can learn to quickly simplify these expressions with ease. There are two common operations that we will work with that involve either multiplication or division.

**Product of Powers** - When you are multiplying two exponential values that share a common base or coefficient the product simply just requires that you find sum of the exponents. It follows the form:

a^{b} x a^{c} = a^{(b + c)}

This applies whether the values of the exponents or fraction. They just need to be rational numbers. We can use this in the example: 4^{ ½} x 4^{ ¼}

We can rewrite this as: 4^{ ½ + ¼} or 4^{¾}.

**Quotient of Powers** - As is often true of division, it serves as the counterpart or polar opposite of multiplication. When we are determining the product, we added the exponents. When we are determining the quotient, we do the polar opposite, we subtract the exponents. This can be illustrated by the backbone:

a^{b} ÷ a^{c} = a^{(b - c)}

Using the same example, but flipping the operation we can run this through for you: 4^{ ½} ÷ 4^{ ¼}

We can rewrite this as: 4^{ ½ - ¼} or 4^{¼}.