10.5 Integer exponents and scientific notation

Use the definition of a negative exponent

The Quotient Property of Exponents , introduced in Divide Monomials , had two forms depending on whether the exponent in the numerator or denominator was larger.

What if we just subtract exponents, regardless of which is larger? Let’s consider $\frac{x^2}{x^5}.$

We subtract the exponent in the denominator from the exponent in the numerator.

We can also simplify $\frac{x^2}{x^5}$ by dividing out common factors: $\frac{x^2}{x^5}.$

This implies that $x^{\mathrm{-3}}=\frac{1}{x^3}$ and it leads us to the definition of a negative exponent    .

The negative exponent tells us to re-write the expression by taking the reciprocal    of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.

When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.

We must be careful to follow the order of operations . In the next example, parts ⓐ and ⓑ look similar, but we get different results.

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.