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Understanding the negative exponent rule is crucial for simplifying expressions with negative exponents. According to this rule, for any non-zero number 'a' and integer 'n', the expression with a negative exponent is converted as follows: \( a^{-n} = \frac{1}{a^n} \).
This means that when you encounter a negative exponent, you take the reciprocal of the base and change the exponent to positive.
For example, if you have \( (6x)^{-2} \), you use the rule to convert it to \( \frac{1}{(6x)^2} \).
This step essentially shifts the term from the numerator to the denominator while changing the exponent to positive.
It's a handy way to readjust terms in an equation while maintaining the mathematical balance.

After applying the negative exponent rule, the next step is to simplify the expression.
Begin by expanding the expression inside the parentheses. For \( (6x)^2 \), this means squaring both 6 and x: \( (6x)^2 = 6^2x^2 \).
Next, perform the calculations for each part.
The number 6 squared is 36 (since \( 6^2 = 36 \)). The variable x squared is simply \( x^2 \).
So, \( (6x)^2 \) simplifies to \( 36x^2 \).
These steps ensure that the expression inside the parentheses is fully expanded and that you understand how each part of the term is handled.
This process eliminates any complexities and breaks down the expression into its simplest form.

The final step involves ensuring all exponents in the expression are positive.
Using the expression from the previous step, you now have \( \frac{1}{36x^2} \).
Notice how all exponents in the final answer are positive.
Writing your final answer with positive exponents provides a standardized form that is typically preferred and often required in mathematical practices.
It simplifies communication and clarity when discussing or solving equations.
This standard form ensures that anyone reading or using the expression understands it correctly without the confusion that negative exponents might introduce.