01Bnegativeindices

= Negative Indices =

Recall this pattern:

If we continue this pattern further, we see negative indices: This leads us to the seventh index law:


 * 7th Index Law (Negative Indices) **

... A number with a __**negative**__ index is equal to the reciprocal of the same number with a **__positive__** index: math . \qquad \bullet \quad a^{-3} = \dfrac{1}{a^3} math

... ... This law also arises from the 2nd Index Law (dividing) ... ... ... {When dividing numbers with the same base, subtract the indices}
 * Note: **

Consider math . \qquad \qquad a^3 \div a^6 = a^{-3} math but math . \qquad \qquad \dfrac{a^3}{a^6} = \dfrac{1}{a^3} math hence math . \qquad \qquad a^{-3} = \dfrac{1}{a^3} math


 * Example 1 **

... ... Write as a fraction in simplest form (without using a calculator)

... ... ** 1) ** . 2 –3 math \\ . \qquad \quad = \dfrac{1}{2^3} \\ . \\ . \qquad \quad = \dfrac{1}{8} math

... ... ** 2) ** . 6 2 × 3 –3 math \\ . \qquad \quad = \dfrac{6^2}{3^3} \\ . \\ . \qquad \quad = \dfrac{36}{27} \\ . \\ . \qquad \quad = \dfrac{4}{3} math


 * Simplifying Indices involving Negative Indices **


 * All the index laws apply to numbers with a negative indices.
 * When simplifying indices, standard practice is to leave the answer with positive indices.


 * Example 2 **

... ... Simplify, leave answer with positive indices.

... ... ** 1) ** math \\ . \qquad \quad a^5b^{-7} \times a^{-2}b^3 \\ . \\ . \qquad \qquad = a^3b^{-4} \\ . \\ . \qquad \qquad = \dfrac{a^3}{b^4} math

... ... ** 2) ** math \\ . \qquad \quad 10a^{-3}b^2c^{-4} \div 2a^{-4}b^5c^{-4} \\ . \\ . \qquad \qquad = 5a^1b^{-3}c^0 \\ . \\ . \qquad \qquad = \dfrac{5a}{b^3} math

... ... ** 3) ** math \\ . \qquad \quad \dfrac{ \left( 2a^3 \right)^{-4} }{8a^{-2}} \\ . \\ . \qquad \qquad = \dfrac{ 2^{-4}a^{-12}}{2^3a^{-2} } \\ . \\ . \qquad \qquad = \dfrac{1}{2^7a^{10}} math.