Solving mathematical problems involving negative exponents may seem daunting. However, negative exponents are treated much like positive exponents when applying the rules for operations. There is an additional rule that allows us to change the negative exponent to a positive one in the denominator of a fraction, and it holds true for any real numbers $n$ and $b$, where $b \neq 0$:

 $b^{-n} = \dfrac{1}{b^n}$

For example:

 $6^{-2} = \dfrac{1}{6^2} = \dfrac{1}{36}$

To understand how this rule is derived, consider the following fraction: 

$\dfrac{7^3}{7^5}$

We can rewrite this as: 

$\dfrac{7 \cdot 7 \cdot 7}{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}$

We then notice that three 7s cancel from both the numerator and denominator, and we are left with: 

$\dfrac{1}{7 \cdot 7} = \dfrac{1}{7^2}$

Note that if we apply the rule for division of numbers with exponents, we have:

$\dfrac{7^3}{7^5} = 7^{(3-5)}= 7^{-2}$

Thus, we can identify that: 

$\dfrac{1}{7^2} = 7^{-2}$

This rule makes it possible to simplify expressions with negative exponents.

Note that each of the rules for operations on numbers with exponents still apply when the exponent is a negative number. For example, consider the rule for multiplying two exponential expressions with the same base. The following is true: 

$3^{-4} \cdot 3^2 = 3^{-4+2} = 3^{-2} = \dfrac{1}{3^2}$

Example

Simplify the following expression: $(2^{-4})^2$.

Note that the rule for raising an exponential expression to another exponent can be applied: 

$(2^{-4})^2 = 2^{(-4)(2)} = 2^{-8} $

This can be simplified using the rule for negative exponents:

 $2^{-8}=\dfrac{1}{2^8}$

Example 

Simplify the following expression: $(3^{-2} \cdot 3^4)^{-3}$.

Recall that the rule for multiplying two exponential expressions with the same base can be applied. Therefore, we can simplify the expression inside the parentheses: 

$3^{-2} \cdot 3^4 = 3^{-2+4} = 3^2$ 

Now place this value back into the parentheses, and apply the rule for raising an exponential expression to an exponent:

$(3^2)^{-3} = 3^{(2)(-3)} = 3^{-6} = \dfrac{1}{3^6}$