Positive Indices
Simplify the following, giving your answers in positive indices only.
Example 1
$$ x^3 × x^{-6} $$
Solution
$$ \because a^m × a^n = a^{m+n} \; \textup{and} \; a^{-n} = \frac{1}{a^n}$$
$$ \therefore x^3 × x^{-6} $$
$$ = x^{3+(-6)} $$
$$ = x^{3-6} $$
$$ = x^{-3} $$
$$ = \frac{1}{x^3} $$
Example 2
$$ b^4 × b^{-9} $$
Solution
$$ \because a^m × a^n = a^{m+n} \; \textup{and} \; a^{-n} = \frac{1}{a^n}$$
$$ \therefore b^4 × b^{-9} $$
$$ = b^{4+(-9)} $$
$$ = b^{4-9} $$
$$ = b^{-5} $$
$$ = \frac{1}{b^5} $$
Example 3
$$ c^6 × c^{-7} $$
Solution
$$ \because a^m × a^n = a^{m+n} \; \textup{and} \; a^{-n} = \frac{1}{a^n}$$
$$ \therefore c^6 × c^{-7} $$
$$ = c^{6+(-7)} $$
$$ = c^{6-7} $$
$$ = c^{-1} $$
$$ = \frac{1}{c^1} $$
$$ = \frac{1}{c} $$