# Properties of Integer Exponents Worksheets

When we are working with exponents, I find that students have a tendency to convert those values to whole numbers before they go any further. It really does not matter if they are middle school, high school, or even college students. This most likely do to learning the concept of PEMDAS aggressively, when they were younger. The habit that I try to get students in is to evaluate all the terms in any equation or expression you are working each and every step of the way. If students do this with operations that include exponents, they will make their lives so much easier along the way. If you understand the properties of integer exponents, three steps and instantly become one. If you need a refresher on all these properties, check the lesson and the bottom of this page. This collection of worksheets and lessons help students understand how to work with exponents in a variety of different scenarios.

### Aligned Standard: Grade 8 Expressions and Equations - 8.EE.A.1

- Multiply Exponents Step-by-Step Lesson- Not only that! There are negative exponents. They're on fire while a wake boarder jumps them! Too over the top?
- Guided Lesson - It's all about simplifying exponents. There is also some division and multiplication of exponents along the way too.
- Guided Lesson Explanation - The division problems make really small numbers and multiplication just the opposite.
- Independent Practice - This work sheet has appeared at two National Conferences already. Here I am just thinking that the bat is cute!
- Matching Worksheet - I know that this can be a tough sheet for some kids. They often get choice "c" and "d" confused. Heads up!
- Exponents to Numbers Five Pack - Put the exponents in numerical form and then compare the values of the exponents.

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

### Homework Sheets

Multiplication and division of exponents followed by exponents of exponents.

- Homework 1 - This is a base with a negative exponent. We can turn it into a positive exponent by using a little math ninja move.
- Homework 2 - If the base is in (a
^{x})^y form, the two exponents should be multiplied. - Homework 3 - When we have the same base to be multiplied with different exponents a^x times a^y, the exponents must be added.

### Practice Worksheets

We follow the same progression in the practice sheets as we did with the homework.

- Practice 1 - Simplify the expressions.
- Practice 2 - Remember what to do with zero and one as exponents.
- Practice 3 - Remember when you should add and subtract exponents.

### Math Skill Quizzes

Once again, we stick with the progression. The quiz problems are standard questions that you will see often.

- Quiz 1 - We bring on quotients into the mix.
- Quiz 2 - Number four is a tick question.
- Quiz 3 - Double exponents you say?

### Rules of Integer Exponent Operations

When we are working with exponents with operations, they can be tricky and often we forget simple rules that can be applied to help us out. There are five surefire ways to speed up your ability to process operations when exponents are involved. Here the rules that students need to follow in regard to the integer exponent operations along with some of the examples.

**Zero-Exponent Rule** - a ^{0} = 1, this rule states that any number raised to the power of zero is equal to 1. This holds true regardless of how large the base number is.

**Product Rule** - This rule states that when two exponents are being multiplied with the same base, the base remains the same as the powers are added together. For example:

a ^{b} × a^{c} = a^{b + c} This would look like this:

9^{5} × 9^{3} = 9^{5 + 3} = 9^{8}

**Quotient Rule** - This rule states that when two exponents of the same base are divided with each other, the base remains the same; the powers of the two are subtracted with each other. Just like division is the opposite of multiplication, the quotient rule is the opposite of the product rule. It is very much familiar to the fraction reduction. When the powers are subtracted, the answers are placed in the denominator or the numerator, depending on where the higher power has been located.

a ^{b} ÷ a^{c} = a^{b - c} This would look like this:

4^{5} ÷ 4^{3} = 4^{5 - 3} = 4^{2}

**Power Rule** - This rule is also called power to power rule: (am)n = amn. According to this rule, when you raise a power to another power, you have to multiply the exponents. Many other rules are associated with the power rule. For example, product-to-powers rule or the quotient to power rule.

(a ^{b})^{c} = a^{(b × c)} This would look like this:

(5 ^{3})^{2} = 5^{(3 × 2)} = 5^{(6)}

**Negative Exponent Rule** - This rule states that negative components, when shifted their position from denominator to numerator, change their sign. If the sign is negative in the denominator, then they turn positive when they become numerator. Here is an example of putting this into action:

x ^{-5} = 1/ x ^{5}