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The law of exponents is a set of rules that describe how to handle mathematical operations involving exponents. One of the most important rules in the context of simplifying expressions, like the one we are discussing, is the division law of exponents. This rule states that when you divide two exponential expressions that have the same base, you subtract the exponent of the denominator from the exponent of the numerator. Consider the base "x" in the expression \(x^a / x^b\). According to the division rule, you simply calculate \(x^{a-b}\). In our problem, dividing \(x^3\) by \(x^7\) gives \(x^{3-7}\), which further simplifies to \(x^{-4}\). This law is extremely useful, enabling us to elegantly simplify expressions by reducing potentially complex divisions to a simple subtraction problem, making calculations more straightforward and manageable.

Exponents can sometimes be negative, and understanding what negative exponents mean is crucial for simplifying expressions. A negative exponent indicates that the base should be placed in the reciprocal form. Simply put, \(x^{-n} = 1/x^n\). So, when faced with \(x^{-4}\), remember it is the same as writing \(1/x^4\). The negative exponent shows that instead of multiplying x by itself, you divide one by \(x\) raised to the positive of that exponent. This method is consistent across any base and exponent, providing a handy tool to transform and simplify expressions. Recognizing and converting negative exponents is frequently one step towards achieving a simpler expression.

When given a fraction involving coefficients, like in our exercise, simplifying them is critical to streamlining the expression. Coefficients are the numerical components of an expression. For example, in \(24/32\), the numbers 24 and 32 are coefficients. The goal is to find the greatest common divisor (GCD) of these coefficients. In this case, the GCD of 24 and 32 is 8. Divide both numbers by their GCD to simplify the fraction:

• 24 divided by 8 is 3
• 32 divided by 8 is 4

Thus, \(24/32 = 3/4\).Creating smaller numbers in the expression makes further operations like applying the law of exponents more straightforward and less error-prone. This simplification step ensures a neat and tidy outcome which is much easier to understand and verify.