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The law of exponents is a crucial mathematical rule that helps us simplify expressions involving powers of the same base. When you encounter division problems like the one in the exercise, where both terms have exponents with the same base, you can apply this law easily.

According to the law, if you divide two terms with the same base, you simply subtract the exponent of the denominator from the exponent of the numerator. This can be expressed as:

• \(a^m \div a^n = a^{m-n}\)

Understanding this fundamental rule allows us to handle complex expressions by reducing them into a simpler form, streamlining our path to the solution. For example, when applying this to \(10^{-5} \div 10^{2}\), we subtract 2 from -5, giving us 10 raised to the power of -7.

Exponents with positive values are often easier to interpret and work with, especially in practical applications like science and engineering. Typically, mathematicians and scientists prefer not to see negative exponents because they involve fractions or reciprocals, which can complicate calculations.

To convert a negative exponent into a positive one, we use the reciprocal of the base. For instance, if you have \(10^{-7}\), it can be rewritten as \(\frac{1}{10^7}\). Here is a step-by-step approach:

• For \(a^{-n}\), rewrite it as \(\frac{1}{a^n}\).
• This transformation gives us a clearer understanding of the scale, as it's often easier to conceptualize numbers greater than 1.

Positive exponents represent how many times you multiply the base by itself, facilitating easier interpretation and computation.

Dividing exponents that share the same base follows a specific rule, making the process straightforward if the base remains constant. This practice leverages the law of exponents for division, as previously discussed.

In the context of functions like \(10^{-5} \div 10^{2}\), we start by ensuring the bases are identical; both bases in this exercise are 10. Then, apply the rule of subtraction to the exponents. In this case, subtracting 2 from -5 yields -7, demonstrating the rule:

• \(a^m \div a^n = a^{m-n}\) translates to \(10^{-5-2} = 10^{-7}\).

This subtractive approach simplifies complex exponent expressions. Finally, rewriting the result with a positive exponent such as \(\frac{1}{10^7}\) ensures clarity and accuracy, especially when interpreting results in broader calculations.