What are exponents and negative exponents?
An exponent is a number that helps show how many times to multiply a base number times itself. When the exponent is negative, it tells us how many times we need to multiply the reciprocal of the base by itself. The rule for negative exponents is:
For example:
In this example, the negative exponent shows that the base x should be in the denominator in order to be positive.
How to use negative exponents:.
Negative exponents tell us how many times we need to multiply the reciprocal of the base by itself.
To make the negative exponent positive, we use the rule:
When the base has a negative exponent in the numerator:
If the base has a negative exponent and is in the numerator, simply write the reciprocal of the base by bringing the base and the exponent down to the denominator. This will change the negative exponent to a positive exponent.
When the base has a negative exponent and is in the denominator:
If the base has a negative exponent in the denominator, write the reciprocal of the base by bringing the base and the exponent to the numerator. The exponent will now be positive.
Note: When moving a base and its negative exponent to the other side of the fraction, make sure to only move the base and exponent. The coefficient will not move, unless they are in parenthesis and raised to the negative exponent.
Example:
The coefficient did not move with the base and exponent.
If the base is not a variable, follow the rule for negative exponents. Find the reciprocal of the base and then multiply the base by itself the amount of times stated by the exponent to get your final answer. In the example below, first find the reciprocal to make the exponent positive.
The reciprocal will be 5³.
The exponent is 3. This shows us we need to multiply the base times itself 3 times.
5 x 5 x 5 = 125
The final answer is 125.
Note: If you have an expression with both negative exponents and positive exponents, you only change the negative exponent to make it positive. The positive exponent does not need to change because it is already positive.
Sample Math Problems
1. Compute
To solve this we need to first write the reciprocal of 2⁻⁴ to make the exponent positive. The reciprocal will be:
To solve this we need to first write the reciprocal of 2⁻⁴ to make the exponent positive. The reciprocal will be:
Now, you need to multiply the base times itself four times because of the exponent 4.
2x2x2x2= 16
The final answer is:
2. Compute
Solution: To solve this first distribute the -5 exponent to both the numerator and denominator. This will give you 1⁻⁵ divided by 5⁻⁵.
Next find the reciprocal because both the numerator and denominator have a negative exponent. The reciprocal will be:
Now multiply each base by their exponent.
5x5x5x5x5=3,125
1x1x1x1x1=1
The final answer will be:
