Negative Exponents (Definition, Rules & Examples)

What is Negative exponents?

Negative exponents are a type of exponential notation and represent the reciprocal of an expression with a positive exponent.

For example, the expression “a^-n” represents the reciprocal of “aⁿ“.

In other words, if “aⁿ” = x, then “a^-n” = 1/x.

For example, if a² = 9, then a^-2 = 1/9.

Some other examples of Negative Exponents:

• 6^-1 is equal to 1/6
• Y^-4 is written as 1/y⁴
• (3x+2y)^-2 is equal to 1/(3x+2y)².

Negative exponents are used to simplifying expressions and to represent the inverse of exponential functions. They can also be used to represent very small numbers in scientific notation.

It is important to comprehend the laws for simplifying expressions with negative exponents, such as:

1. Law of quotient 

When dividing two exponential expressions with the same base, the result is the difference between the exponents.

For example, a^m / aⁿ = a^{m – n}.

2. Fractions with Negative Exponents

If a^m is an exponential expression, then we can express:

(a^-m )^{(1/n) =}  (1/a^m)^(1/n).

For example, (a^-2)^(1/3) = (1/a^2)^(1/3).

By using these rules, you can simplify expressions with negative exponents and manipulate them to solve mathematical problems.

How to Solve Negative Exponents?

Here are some examples of solving the negative exponents.

Example 1: Simplify 6x^-2.

Solution:

As Given expression 6x^-2

Using the rule, a^-n = 1/aⁿ

6x^-2 = 6 (1/x²)

6x^-2 = 6/x².