# Negative Exponents Worksheet, Rules, and Examples

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### Key Points about Negative Exponents

- Negative exponents represent the inverse of a number raised to a positive exponent.
- Negative exponents are crucial for solving complex equations involving variables and exponents.
- Negative exponents are used in scientific notation to represent very small numbers.

## Negative Exponents Explained

**Negative Exponents** refer to bases that are raised to a power that is negative. In order to simplify bases raised to **negative exponents**, you must make the exponents positive. The shortcut for changing **negative exponents** into positive exponents, is to flip the term with a **negative exponent** over the fraction line. Once the term has been flipped over the fraction line, the exponent becomes positive.

If there is nothing left on one side of the fraction line, you use the number one as a place holder. If there are any other like terms, you must simplify the problem by either multiplying or dividing the exponents of the like terms.

Negative exponents can be a challenging concept to grasp for many students, but they are an essential part of algebra and higher-level math. In simple terms, negative exponents represent the inverse of a number raised to a positive exponent. For example, 2^-3 is the same as 1/(2^3), or 1/8.

Understanding negative exponents is crucial for solving complex equations involving variables and exponents. It is also essential in scientific notation, where negative exponents are used to represent very small numbers. While negative exponents may seem intimidating at first, with practice, they can become second nature.

**Common Core Standard: **8.EE.A.1**Related Topics: **Product Rule, Quotient Rule, Power of a Power Rule, Power of a Quotient Rule, Power of a Product Rule**Return To: **Home, 8th Grade

## What do Negative Exponents Mean?

Negative exponents are a way of representing fractions or decimals in a more compact form. A negative exponent indicates how many times the base should be divided by itself.

For example, if the exponent is -3 and the base is 2, then 2 raised to the power of -3 is equal to 1 divided by 2 raised to the power of 3. This can be simplified to 1/8.

In general, if a number is raised to a negative exponent, it is the reciprocal of the same number raised to the positive exponent. For instance, if 3 is raised to -2, then it is equivalent to 1/3 raised to 2. This is because 3 raised to -2 is equal to 1 divided by 3 raised to 2.

Negative exponents can be used to represent very large or very small numbers. For instance, instead of writing 0.0001, one can write 10 raised to the power of -4. Similarly, instead of writing 1000000, one can write 10 raised to the power of 6.

It is important to note that negative exponents only apply to powers of numbers, not to the numbers themselves. For example, -2 raised to the power of 3 is equal to -8, but (-2) raised to the power of 3 is equal to -8.

In summary, negative exponents are a shorthand way of writing fractions or decimals. They represent the reciprocal of the base raised to the positive exponent. They can be used to represent very large or very small numbers.

## How to Solve Negative Exponents

Negative exponents can be intimidating, but they don’t have to be. With a little bit of practice, anyone can learn how to solve them. This section will cover the basics of multiplying and dividing negative exponents, simplifying expressions with negative exponents, and working with fractional exponents.

### Multiplying and Dividing

When multiplying or dividing numbers with negative exponents, remember that a negative exponent means the reciprocal of the number. For example, if you have `2^-3`

, that is the same as `1/2^3`

. Similarly, if you have `1/3^-2`

, that is the same as `3^2`

.

To multiply numbers with negative exponents, first multiply the base numbers together, then add the exponents. For example, `2^-3 * 3^-2`

is equal to `1/2^3 * 1/3^2`

, which simplifies to `1/18`

.

To divide numbers with negative exponents, first divide the base numbers, then subtract the exponents. For example, `2^-3 / 3^-2`

is equal to `1/2^3 / 1/3^2`

, which simplifies to `9/8`

.

### Simplifying Expressions

When simplifying expressions with negative exponents, remember to follow the order of operations. Start by simplifying any expressions inside parentheses, then work from left to right, simplifying any exponents, multiplication, and division before addition and subtraction.

For example, `(2^-3 * 3^-2)^2`

can be simplified as follows:

```
(2^-3 * 3^-2)^2
= (1/2^3 * 1/3^2)^2
= (1/8 * 1/9)^2
= 1/5184
```

### Fractional Exponents

Fractional exponents are a way of expressing roots. For example, `2^(1/2)`

is the same as the square root of 2. When dealing with fractional exponents, remember that the denominator of the fraction is the root and the numerator is the exponent.

To simplify expressions with fractional exponents, first simplify any expressions inside parentheses, then work from left to right, simplifying any exponents, multiplication, and division before addition and subtraction. For example, `(2^(1/2))^3 * 2^(1/2)`

can be simplified as follows:

```
(2^(1/2))^3 * 2^(1/2)
= (2^(3/2)) * 2^(1/2)
= 2^(3/2 + 1/2)
= 2^2
= 4
```

Remember that negative exponents can be a bit tricky, but with practice and patience, anyone can learn how to solve them.

## Negative Exponents with Variables

Negative exponents with variables can be simplified using the same rules as negative exponents with numbers. The exponent rule states that any base with a negative exponent can be written as its reciprocal with a positive exponent.

For example, if x is a variable, then x^-2 can be written as 1/x^2. Similarly, if y is a variable, then y^-3 can be written as 1/y^3.

When working with negative exponents with variables, it is important to remember that the exponent only applies to the variable and not any coefficients that may be present. For instance, in the expression 2x^-3, the exponent -3 only applies to x and not 2. Therefore, 2x^-3 can be simplified as 2/x^3.

To simplify expressions with negative exponents with variables, the following steps can be followed:

- Identify the base and exponent of the negative exponent.
- Write the reciprocal of the base with a positive exponent.
- Simplify any coefficients or constants that may be present.

For example, consider the expression (3x^-2)/(2y^-3). To simplify this expression, we can follow the steps below:

- The base of the first negative exponent is x and the exponent is -2. The base of the second negative exponent is y and the exponent is -3.
- The reciprocal of x^-2 is x^2 and the reciprocal of y^-3 is y^3.
- Simplify the coefficients to get (3/2) * (x^2/y^3).

In summary, negative exponents with variables can be simplified using the same rules as negative exponents with numbers. It is important to remember that the exponent only applies to the variable and not any coefficients that may be present.

**3 Simple Negative Exponents Examples**

Negative exponents occur when the exponent of a number is a negative integer. In this case, the base is raised to the reciprocal of the absolute value of the exponent.

- Whenever a term has a negative exponent, it is not simplified.
- You must make all of the negative exponents positive.
- Flip any term with a negative exponent over the fraction bar.
- When you flip the term over the negative exponent, you make the exponent positive.

Here are some examples that illustrate how negative exponents work:

### Example 1

Consider the expression 2^-3. Here, the base is 2 and the exponent is -3. To evaluate this expression, take the reciprocal of the base and raise it to the absolute value of the exponent. Thus, 2^-3 = 1/2^3 = 1/8.

### Example 2

Let’s say you have the expression (3/4)^-2. In this case, the base is 3/4 and the exponent is -2. To evaluate this expression, take the reciprocal of the base and raise it to the absolute value of the exponent. Thus, (3/4)^-2 = (4/3)^2 = 16/9.

### Example 3

Suppose you have the expression 5^-1/2. Here, the base is 5 and the exponent is -1/2. To evaluate this expression, take the reciprocal of the base and raise it to the absolute value of the exponent. Thus, 5^-1/2 = 1/5^1/2 = 1/√5.

### Practice Problems

Here are some practice problems to help you understand negative exponents better:

- Simplify 2^-4.
- Evaluate (1/3)^-2.
- Simplify (5/6)^-3.

By solving these problems, you can become more comfortable with negative exponents and their applications in mathematics.

## 5 Quick Negative Exponents Practice Problems

## Negative Exponent Rule

The Negative Exponent Rule is a fundamental rule of exponents that states that for any non-zero real number `a`

and any integer `n`

, `a^-n`

is equal to `1/a^n`

or `1/(a^n)`

. In other words, raising a number to a negative exponent is the same as taking the reciprocal of the number raised to the corresponding positive exponent.

For example, `2^-3`

is equal to `1/(2^3)`

or `1/8`

. Similarly, `x^-2`

is equal to `1/(x^2)`

. The Negative Exponent Rule is a critical component of the Laws of Exponents, which are a set of rules that govern the manipulation of exponents in algebraic expressions.

It is essential to note that leaving negative exponents in an answer is considered poor form in mathematics. All answers should always be simplified to show positive exponents.

The Negative Exponent Rule can be extended to apply to variables and expressions with multiple terms. For instance, `(ab)^-3`

is equal to `1/(ab)^3`

or `1/(a^3b^3)`

. Similarly, `(x^2y^-3)^-2`

is equal to `1/(x^2y^-3)^2`

or `1/(x^4y^-6)`

, which simplifies to `y^6/x^4`

.

In summary, the Negative Exponent Rule is a fundamental rule of exponents that states that raising a number to a negative exponent is the same as taking the reciprocal of the number raised to the corresponding positive exponent. It is a critical component of the Laws of Exponents and should be used to simplify algebraic expressions.

## Negative Exponents Fractions

Negative exponents with fractions can be tricky, but with a little practice, they can be simplified easily. A negative exponent tells us how many times to divide the base number. For example, 2^-3 means dividing 1 by 2 three times, which gives us 1/2^3 or 1/8.

When dealing with fractions, we use the rule that states that a negative exponent in the denominator becomes a positive exponent in the numerator (and vice versa). For instance, 2^-2/3 is the same as 1/2^(2/3).

Multiplying fractions with negative exponents can be simplified by first converting the negative exponents to positive exponents. For example, (2/3)^-2 * (3/4)^-3 can be simplified to (3/2)^2 * (4/3)^3.

It’s important to note that negative exponents can also be written as fractions with a numerator of 1. For example, 2^-3 can be written as 1/2^3. This form can be helpful when dealing with complex fractions that have negative exponents.

Here is a table summarizing the rules for negative exponents with fractions:

Expression | Simplified Form |
---|---|

a^-n/b | b^n/a^n |

(a/b)^-n | (b/a)^n |

a^-n | 1/a^n |

Remember, negative exponents with fractions can be simplified by converting them to positive exponents or fractions with a numerator of 1. With practice, simplifying expressions with negative exponents and fractions can become second nature.

## Negative Exponents FAQ

### What is the rule for negative exponents?

The rule for negative exponents states that for every number “a” with negative exponents “-n” (i.e.), a^-n, take the reciprocal of the base number and multiply the value according to the value of the exponent number. For example, 4^-3 is equal to 1/(4^3).

### How do you solve for a negative exponent?

To solve for a negative exponent, take the reciprocal of the base number and change the sign of the exponent. For example, 2^-4 is equal to 1/(2^4) or 1/16.

### What is 10 to the negative power of 2?

10 to the negative power of 2 is equal to 1/100 or 0.01.

### What is the negative rule example?

The negative rule example is that when a negative exponent is present, the base number is moved to the denominator of the fraction. For example, 2^-3 is equal to 1/(2^3) or 1/8.

### What is the difference between the positive and negative exponents?

The difference between positive and negative exponents is that positive exponents indicate multiplication while negative exponents indicate division. For example, 2^3 is equal to 2 x 2 x 2 or 8, while 2^-3 is equal to 1/(2^3) or 1/8.

### Mention the rules for negative exponents?

The rules for negative exponents are: (1) any number raised to the power of zero is equal to 1, (2) any number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent, and (3) when dividing numbers with the same base, subtract the exponents.

### What is an example of a negative exponent in real life?

An example of a negative exponent in real life is measuring the concentration of a substance in a solution. The concentration is often expressed in terms of moles per liter, which is written as mol/L. If the concentration is very small, it may be expressed as a negative exponent, such as 1 x 10^-6 mol/L.

### How do you simplify negative exponents outside parentheses?

To simplify negative exponents outside parentheses, first distribute the exponent to each term inside the parentheses. Then, take the reciprocal of each term and change the sign of the exponent. Finally, simplify the expression by combining any like terms. For example, (2^-3)(3^-2) is equal to (1/(2^3))(1/(3^2)) or 1/72.

## Negative Exponents Worksheet Video Explanation

Watch our free video on how to solve **Negative Exponents**. This video shows how to solve problems that are on our free **Negative Exponent **worksheet that you can get by submitting your email above. The **Negative Exponents Rules** shown in the video will work on any problem that involve Negative Exponents.

**Watch the free Negative Exponents video on YouTube here: Negative Exponents Worksheet**

**Video Transcript:**

This video is about using the negative exponent rule. We’re going to use our negative exponent rule worksheet to show you a couple example problems showing you how to do negative exponents.

Here is the first problem in our negative exponents practice worksheet. We are given five to the negative ninth. Now in order to simplify each expression with a negative exponent you have to make the exponent positive. In the case of number one this exponent here of negative nine needs to be turned into a positive nine. The way you make this exponent positive is that you have to flip it on to the other side of the fraction bar.

That might not make any sense at first but you have to remember that all whole numbers are written over 1. 5 to the negative ninth actually has a 1 or is being divided by one underneath of it, even though we don’t write it every single time. In order to make this exponent positive you have to take your term here, which is 5 to the negative ninth, and you have to move the entire thing to the bottom of the fraction bar. After you do that the exponent will become positive.

We take 5 to the negative ninth, we can write our fraction bar. If we want it becomes 5 you keep the base and then you make the exponent positive 5 to the positive ninth our term on top is gone but we have to include a placeholder. The placeholder we’re going to use is the number one because one divided by five to the ninth doesn’t change the actual value of the term. This will be our answer and it shows how to make a negative exponent positive.

Moving on to number three on our negative exponents rule we have 1 over 7 to the negative twentieth. Now the same process we used in number one also works in number three except instead of moving our term to the bottom, like we did, our term is already on the bottom. We’re going to move it to the top what we’re trying to do is we’re trying to make the exponent positive. This negative 20 needs to turn into a positive 20. We’re going to take our term 7 to the negative 20th and we’re going to move it to the top of this term, to the top of the fraction bar.

It will become seven to the positive twentieth. Anytime you flip a number or term over the fraction bar sign on the exponent changes. Now our negative twenty becomes positive and then on bottom we have to use a placeholder to keep the number in correct numerical terms. And we’re going to use one now seven to the 20th power divided by one is just seven to the 20th power and that’s our answer.

The last example for the negative power rule that we’re going to do is a little bit trickier we’re given X to the negative third divided by X to the fourth. The first thing we need to do is we need to get rid of this negative three. We have to make it positive, so just like in the other examples, the way you do that is you flip it over the fraction bar. We will write X to the fourth because it’s on the bottom so it stays on the bottom. We don’t change anything about that we just moved it over and now we have also X to the positive third, which now has been moved to the bottom. We took our term here we move the whole thing to the bottom and we rewrote it right next to X to the fourth. Now on top we have to use 1 as a placeholder.

The next step is we have to simplify X to the 4 times X to the 3rd. When we do that we’re just going to multiply them together. And you will remember that when you multiply terms with exponents you add the exponents. We’ll keep the base of X and then we’ll do 4 plus 3 which is 7 and 1 over X to the positive 7th is our solution. You can try the practice problems by downloading the free negative exponents worksheets above.

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