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# Negative Exponents

In math,
exponents
define the number of times a number is multiplied by itself. They
are written with a base number like this:
${4}^{3}$

where 4 is the base number and 3 is the exponent. The previous term
is read as "4 to the 3rd power" and indicates that 4 is multiplied
by itself 3 times, or
$4\times 4\times 4$

An exponent can be positive, negative, or zero, but they are handled
differently.

## Negative exponent definition

We know that a positive exponent tells us to multiply a number by
itself a particular number of times. A negative exponent, on the
other hand, tells us how many times to divide the base number. In
practice, the negative exponent means that we have to multiply the
reciprocal of the base a certain number of times.

An example of a negative exponent is
${4}^{-2}$

We can write
${4}^{-2}$
as
${\frac{1}{4}}^{2}$

The value of
${4}^{-2}$
is therefore
$\frac{1}{16}$
because
$\frac{1}{4}\times \frac{1}{4}=\frac{1}{16}$

## Rules of negative exponents

There are two important rules of negative exponents that simplify
solving problems involving them.

**Negative Exponent Rule 1**

For every number "a" with a negative exponent, "-n", multiply the
value of the reciprocal of the base number according to the value of
the exponent number.

For example,
${6}^{-3}$
. The base number is 6 and the exponent is -3.

According to the rule,
${6}^{-3}$
is written as
$\frac{1}{{6}^{3}}$
, which is written as "
$\frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}$
", which equals
$\frac{1}{216}$
.

This means the value of
${6}^{-3}=\frac{1}{216}$
.

**Negative Exponent Rule 2**

For every number "a" in the denominator with a negative exponent
"-n", the result can be written in the form of
$a\times a\times a$
.. according to the exponent.

As an example, take the fraction
$\frac{1}{{3}^{-3}}$
.

This time, the negative exponent is in the denominator of the
fraction.

Here,
$\frac{1}{{3}^{-3}}$
can be rewritten as , which is equal to
$3\times 3\times 3$
, which equals 27.

Therefore,
$\frac{1}{{3}^{-3}}=27$
.

## Using the product of powers property to show how negative exponents work

You can use the
product of powers property
to show how negative exponents work as follows.

Here's the plain text representation of the given equations:

${7}^{-2}\times {7}^{2}={7}^{-2+2}={7}^{0}$

We know that
${7}^{2}$
equals 49, and we know that
${7}^{0}$
equals 1. This tells us that
${7}^{-2}\times 49$
equals 1.

What number times 49 equals 1? That would be its multiplicative
inverse,
$\frac{1}{49}$
.

Therefore,
${7}^{-2}$
equals
$\frac{1}{49}$
.

The general rule for this is written as:

For all
real numbers
a and b where
$a\ne 0$

${a}^{-b}$
is defined as
$\frac{1}{{a}^{b}}$
,

$\frac{1}{{a}^{-b}}$
is defined as
${a}^{b}$
.

## Topics related to the Negative Exponents

## Flashcards covering the Negative Exponents

## Practice tests covering the Negative Exponents

## Get help learning about negative exponents

Tutoring is a great way for your student to supplement their
in-class learning about negative exponents. A tutor can clarify
anything the teacher has explained that your student has had a
difficult time understanding. By learning your student's learning
style, their tutor can customize lessons to cater to the way their
mind works, making them more effective. A tutor can answer your
student's questions as soon as they come up so they are able to
solve problems correctly right from the start. If you'd like to
learn more about how tutoring can help your student understand
negative exponents and other mathematical topics, contact a helpful
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