Define a negative exponent.

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.

In more detail, when you see a negative exponent, it means you need to take the reciprocal (or the "flip") of the base number and then raise it to the positive version of that exponent. For example, if you have \(a^{-n}\), this is equivalent to \(\frac{1}{a^n}\). So, \(2^{-3}\) would be \(\frac{1}{2^3}\), which simplifies to \(\frac{1}{8}\).

Understanding negative exponents is crucial because they often appear in various mathematical contexts, such as algebra and scientific notation. They help simplify expressions and solve equations more efficiently. For instance, in scientific notation, a number like \(3.2 \times 10^{-4}\) represents \(3.2 \times \frac{1}{10^4}\), which is \(0.00032\).

When dealing with negative exponents, remember the key rule: \(a^{-n} = \frac{1}{a^n}\). This rule applies to any non-zero base \(a\) and any integer \(n\). It’s also important to note that this rule helps maintain the consistency of the laws of exponents, such as the product of powers and the power of a power rules.

By practising problems involving negative exponents, you’ll become more comfortable with these concepts and better prepared for more advanced topics in mathematics.