# Negative exponents

Here you will learn about negative exponents, including how to simplify and evaluate with negative exponents.

Students will first learn about negative exponents as part of expressions and equations in 8 th grade, and will continue to expand their knowledge through high school.

## What are negative exponents?

Negative exponents are powers (also called indices) with a negative sign (minus sign) in front of them.

Examples of negative exponents:

• x^{-2}
• 3^{-4}
• 2b^{-\frac{1}{2}}

You get negative exponents by dividing two terms with the same base where the first term is raised to a power that is smaller than the power that the second term is raised to. Similarly to how a positive exponent means repeated multiplication, a negative exponent means repeated division.

For example,

x^{3} \div x^{4}=\cfrac{x \times x \times x}{x \times x \times x \times x}

When you cancel the common factors of x,

\cfrac{ \cancel{x} \times \cancel{x} \times \cancel{x}}{ \cancel{x} \times \cancel{x} \times \cancel{x} \times x}

You are left with,

\cfrac{1}{x}

Using the division law of exponents, you know that,

x^{3} \div x^{4}=x^{3-4}=x^{-1}

So,

x^{-1}=\cfrac{1}{x}

### How to use negative exponents

A negative exponent can be defined as the multiplicative inverse of the base raised to the power, which is of the opposite sign of the given power.

In other words, in order to make the negative exponent positive, put the term over 1 and flip it. It is known as finding the reciprocal of the base (term).

The negative exponent rule states that a number with a negative exponent should be put in the denominator.

For example,

x^{-2}=\cfrac{x^{-2}}{1}=\cfrac{1}{x^{2}}

\cfrac{1}{x^{2}}

is the same as

x^{-2}

For example,

2^{-3}=\cfrac{2^{-3}}{1}=\cfrac{1}{2^{3}}=\cfrac{1}{8}

Negative exponents will often be used in conjunction with other exponent laws, including division, parentheses, and multiplication laws.

You can also figure out the value of negative number expressions by identifying patterns. For example, notice how when the power decreases by 1, the answer is half of the answer of the previous expression.

\begin{aligned} & 2^{3}=8 \\\\ & 2^{2}=4 \\\\ & 2^{1}=2 \\\\ & 2^{0}=1 \\\\ &2^{-1}=\cfrac{1}{2} \end{aligned}

What do you think 2^{-2} will be equal to?

## Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8: Expressions and Equations (8.EE.A.1)
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 \times 3-5 = 3-3 = \cfrac{1}{33} = \cfrac{1}{27}.

## How to evaluate negative exponents

In order to evaluate a negative exponent, you need to:

1. Put the term over \bf{1}.
2. Flip the fraction to make the exponent positive.
3. Simplify, if necessary.

## Negative exponents examples

### Example 1: no coefficient in front of the base

Simplify and leave your answer in index form.

a^{-4}

1. Put the term over \bf{1}.

When the exponent is negative, put it over 1,

\cfrac{a^{-4}}{1}

2Flip the fraction to make the exponent positive.

Flip and change the power from -4 to +4.

\cfrac{1}{a^{4}}

### Example 2: negative exponents

Simplify and leave your answer in index form.

7^{-9}

Put the term over \bf{1}.

Flip the fraction to make the exponent positive.

Simplify, if necessary.

### Example 3: with a coefficient in front of base

Simplify and leave your answer in index form.

(10 a)^{-3}

Put the term over \bf{1}.

Flip the fraction to make the exponent positive.

Simplify, if necessary.

### Example 4: with a coefficient in front of base

Simplify and leave your answer in index form.

3b^{-2}

Put the term over \bf{1}.

Flip the fraction to make the exponent positive.

### Example 5: with fractional exponents

Evaluate

\left(\cfrac{4}{3}\right)^{-2}

Put the term over \bf{1}.

Flip the fraction to make the exponent positive.

Simplify, if necessary.

### Example 6: with fractional exponents

Evaluate

\left(\cfrac{14}{4}\right)^{-2}

Put the term over \bf{1}.

Flip the fraction to make the exponent positive.

Simplify, if necessary.

### Teaching tips for negative exponents

• Introduce the concept using concrete examples that can illustrate it. Using simple numerical examples can make it easy to show how negative exponents relate to taking the reciprocal.

• Offer a variety of practice problems to reinforce the concept. This can be done using a worksheet, but find an interactive way to allow students to discuss and problem solve together.

• Allow students the opportunity to explore real-life applications of negative exponents in fields such as science, engineering, or finance.

### Easy mistakes to make

• Confusing integer and fractional powers
Raising a term to the power of 2 means you square it.
For example,
2^{2}=2 \times 2
Raising a term to the power of \cfrac{1}{2} means we find the square root of it.
For example,
2^{\frac{1}{2}}=\pm \sqrt{2}

• Thinking indices, powers or exponents are all different
Exponents can also be called powers or indices.

• Not turning a negative exponent into a positive exponent when flipping the term
After putting the term over one, make sure to flip the exponent from a negative to a positive exponent.

• Making a mistake when writing one over a fraction
Typically, when finding the reciprocal, you write the term over one and flip the term. However, when you are finding the reciprocal of a fraction, the steps are a little different. This is why when dealing with fractions it is easier to skip to changing the powers and just flip the fraction.

### Practice negative exponents questions

1. Simplify. Express your answer in index form.

x^{-5}

x^{5}

\cfrac{1}{x^{-5}}

\cfrac{1}{x^{5}}

\cfrac{1}{5x}

The negative exponent means finding the reciprocal, so

x^{-5}=\cfrac{1}{x^{5}}

2. Simplify. Express your answer in index form.

(2 b)^{-4}

16b^{-4}

\cfrac{1}{2 b^{4}}

\cfrac{1}{8 b^{4}}

\cfrac{1}{16 b^{4}}

The negative exponent means finding the reciprocal, so

\begin{aligned} & (2 b)^{-4} \\\\ & =\cfrac{1}{(2b)^{4}} \\\\ & =\cfrac{1}{2b\times2b\times2b\times2b} \\\\ & =\cfrac{1}{16 b^{4}} \end{aligned}

3. Simplify. Express your answer in index form.

3 b^{-3}

\cfrac{1}{b^{3}}

\cfrac{3}{b^{3}}

\cfrac{3}{3b}

\cfrac{1}{3b}

The negative exponent means finding the reciprocal, but this only applies to the variable, so

3 b^{-3}=\cfrac{3}{b^{3}}

4. Evaluate. Express your answer in index form.

\left(\cfrac{5}{8}\right)^{-1}

-\cfrac{8}{5}

\cfrac{8}{5}

-\cfrac{3}{5}

\cfrac{3}{5}

The negative exponent means finding the reciprocal, which means inverting the fraction, so

\left(\cfrac{5}{8}\right)^{-1}=\cfrac{8}{5}

5. Evaluate. Express your answer in index form.

9^{-2}

\cfrac{1}{81}

\cfrac{1}{9^2}

\cfrac{1}{18}

-\cfrac{1}{81}

The negative exponent means finding the reciprocal, so

\begin{aligned}& 9^{-2}=\cfrac{1}{9^2} \\\\ & =\cfrac{1}{9 \times 9} \\\\ & =\cfrac{1}{81} \end{aligned}

6. Evaluate. Express your answer in index form.

\left(\cfrac{4}{5}\right)^{-2}

\left(\cfrac{4}{5}\right)^{2}

\cfrac{16}{25}

\cfrac{25}{16}

-\cfrac{25}{16}

The negative exponent means finding the reciprocal, so

\begin{aligned} &\begin{aligned} & \left(\cfrac{4}{5}\right)^{-2}=\left(\frac{5}{4}\right)^2 \\\\ & =\left(\cfrac{5}{4}\right) \times\left(\cfrac{5}{4}\right) \end{aligned} \\\\ &\begin{aligned} & =\cfrac{5 \times 5}{4 \times 4} \\\\ & =\cfrac{5^2}{4^2} \\\\ & =\cfrac{25}{16} \end{aligned} \end{aligned}

## Negative exponents FAQs

Can you have a negative base with a negative exponent?

Yes, the base can be a positive number, negative number, or zero. The negative exponent only affects the power to which the base is raised to.

How are negative exponents related to scientific notation?

Scientific notation involves negative exponents often. They can represent numbers in the form of a \times 10^{-n}, where a is a number between 1 and 10.

What is the difference between a negative power and a positive power?

A positive power results in the multiplication of the base by itself multiple times, while negative powers result in taking the reciprocal of the base raised to the corresponding positive exponent.

Can a base have a zero exponent?

Yes, according to the properties of exponents, the power of zero, any nonzero base raised to the zero power equals 1.

What are the rules of exponents?

Some of the fundamental rules of exponents are:

∘ Product rule: \left(a^m \times a^n\right)=a^{m+n}

∘ Quotient rule: \cfrac{a^m}{a^n}=a^{m-n}

∘ Power rule: \left(a^m\right)^n=a^{m \times n}

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