We already saw division of exponents two times:
- when discussed fractional exponents (anm=nam)
- when discussed multiplication of exponents (indeed, anm=am⋅n1=(am)n1=nam).
Rule for dividing exponents: nam=anm.
Word of caution. It is not always possible to interchange an exponent and a nth root, i.e. nam=(na)m.
It is always possible for positive number, but not for negative.
Example.
4(−5)2=425=452=542=521=5,
but (4−5)2 even doesn't exists, because 4−5 doesn't exist.
Next couple of examples just show a couple of common problems.
Example 2. Rewrite, using positive exponents: 52−31.
Just apply above rule: 52−31=25−31=2−151.
Now, just rewrite using positive exponent: 2−151=21511.
What if we have a couple of radicals?
Example 3. Simplify: 43251.
We start form innermost number: 435−2=45−32=54−32=5−61=5611=651.
Let's see how interchanging works.
Example 4. Simplify: (3(−2)4)6.
3(−2)46=(316)6=1636=162=256.
Now, it is time to exercise.
Exercise 1. Rewrite, using positive exponents: 623.
Answer: 221=2.
Exercise 2. Find 4(−5)2 and (4−5)2.
Answer: 4−52=5 and (4−5)2 doesn't exist.
Exercise 3. Find 5(−2)3.
Answer: (−2)53.
Exercise 4. Rewrite, using positive exponents: 4527.
Answer: 2207.
Exercise 5. Rewrite, using positive exponents: 53−25.
Answer: −32=−231.