# Evaluating Negative Exponents Worksheets

We have run into exponents before especially when dealing with squared and cubed values. Until now we have often thought that exponents are used to amplify any situation they are brought into. They do help us quantify very large values. Negative exponents, as the name implies, do just the opposite of this. They help us work with very small values and express or share those values. You find negative exponents used by scientists to share very small values. In fact there is specific form of notation called scientific notation that focus just on that aspect. This well written series of worksheets and lessons will teach students how to determine the value of problems that include negative exponents.

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

- Anatomy of Exponents Lesson- We spell it out for you. Positive exponents = multiply and negative = divide.
- Expressions with Negative Exponents Lesson - We show you multiple ways to solve this problem.
- Evaluating Negative Exponents Practice Worksheet 1 - There are two distinct skills that we work on here.
- Simplifying Expressions with Negative Exponents - It's all about viewing the problem from a far.
- Multiplication and Division with Negative Exponents - This is a slightly advanced skill for this level, but this should challenge your top students.

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

### Evaluating Negative Exponents Practice Medium Difficulty

These are just one notch tougher than worksheet 1.

- Homework 2 - Apply the negative exponent rule and simplify.
- Homework 3 - Find the missing value.

### Multiplication and Division with Negative Exponents

These really raise the bar with the advanced operations.

- Practice 2 - Eliminate the negative exponents and simplify. This a some real quality problems.
- Practice 3 - Here is the math Ninja, we have referenced before for you. He slices an dices math problems.

### Upper Level Difficulty

Here is a two-step progression for students.

- Simplifying Advanced Expressions - This is a little more complicated. You will need to break this down in parts.
- Simplifying Expressions - These problems come in stacks. Start by simplifying anything that is possible for you.

### General Rules of Negative Exponents

How do you go about evaluating the value of a negative exponents? Evaluating exponents is one of the most common aspects of basic mathematics. Exponents are also called Indices or Powers. An exponent of any number indicates how many times will that number be multiplied with itself. For example:

8^{2} = 8 x 8 = 64

In other words, 8^{2} can also be called the "8 to the second power", "8 squared" or "8 to the power of 2".

So, what to do when you have negative exponents. For example, something like 8^{-2} What does this mean?

Negative exponents are an inverse of the positive exponents, i.e., in positive exponents, the numbers get multiplied. When negative exponents are involved, it means that division operation will take place. Negative exponents indicate that you must take the multiplicative inverse of the base number. You then raise this number to the opposite of power that you were given.

For example, if we were to evaluate: 5^{-3}

**Step 1) Take the multiplicative inverse of the base.**

We can consider 5 as 5/1 to start. The multiplicative inverse would therefore be: 1/5.

**Step 2) Raise this new base value to the opposite of the given power.**

We were initially given the power -3. The opposite of -3 is 3. We would therefore be left with the value:

(1/5)^{3}

**Step 3) Reduce and simplify**

We can rewrite this value as a fractional multiplication problem such as:

(1/5) (1/5) (1/5).

If you remember fractional multiplication, we find the product of the numerator and place them over the product of the denominators. The numerator product would be: 1 x 1 x 1 = 1. The denominator product would be: 5 x 5 x 5 = 125. Our final value would be: 1/125. We can also write this as the decimal value: 0.008

When working with these types of problems there are some general rules that you should keep under consideration. These rules will help you keep a clear mental picture of what is going on here. When dealing with positive powers we understand that we are just multiplying the base number by itself the number of times indicated by the index. Just like the negative value is the opposite of a positive value, the same is true of the operations that result when dealing with exponents. When dealing with negative exponents the value of the index indicates how many times the base number is divided by itself. The general rule for raising a value to a negative power or index is that the numerator switches places with the denominator of the base value. In most cases this means that you place that same value over one. Which is another way to say that you just take the reciprocal of the base value.