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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 × 4 × 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 1 4 2
The value of 4 -2 is therefore 1 16 because 1 4 × 1 4 = 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 1 6 3 , which is written as " 1 6 × 1 6 × 1 6 ", which equals 1 216 .
This means the value of 6 -3 = 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 × a × a .. according to the exponent.
As an example, take the fraction 1 3 -3 .
This time, the negative exponent is in the denominator of the fraction.
Here, 1 3 -3 can be rewritten as , which is equal to 3 × 3 × 3 , which equals 27.
Therefore, 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 × 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 × 49 equals 1.
What number times 49 equals 1? That would be its multiplicative inverse, 1 49 .
Therefore, 7 -2 equals 1 49 .
The general rule for this is written as:
For all real numbers a and b where a 0
a - b is defined as 1 a b ,
1 a - b is defined as a b .

Topics related to the Negative Exponents

Solving Two-Step Linear Equations
Describing the Graph of a Function
Combining Like Terms with Exponents

Flashcards covering the Negative Exponents

Algebra 1 Flashcards
College Algebra Flashcards

Practice tests covering the Negative Exponents

Algebra 1 Diagnostic Tests
College Algebra Diagnostic Tests

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 Educational Director at Varsity Tutors today.
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