Problem 141 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(\frac{4 m^{-3} n^{6}}{2 m^{-5} n}\right)^{3} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $(\frac{4 m^{-3} n^{6}}{2 m^{-5} n})^{3}$ so that it contains only positive exponents. Answer: $8 m^{6} n^{15}$

Step 1: Apply the exponent rule to the whole expression

To begin simplifying the given expression, let's first apply the exponent rule \((a^m)^n = a^{mn}\) to the whole expression: $$\left(\frac{4 m^{-3} n^{6}}{2 m^{-5} n}\right)^{3} = \frac{(4 m^{-3} n^{6})^3}{(2 m^{-5} n)^3}$$

Step 2: Simplify the numerator and denominator

Now, let's simplify the numerator and denominator separately, applying the exponent rule \((a^m)^n = a^{mn}\) for each term: In the numerator \((4 m^{-3} n^{6})^3\), we have: $$(4^3)(m^{-3\cdot 3})(n^{6\cdot 3})= 64 m^{-9} n^{18}$$ In the denominator \((2 m^{-5} n)^3\), we have: $$(2^3)(m^{-5\cdot 3})(n^3)= 8 m^{-15} n^3$$ So the whole expression now looks like this: $$\frac{64 m^{-9} n^{18}}{8 m^{-15} n^3}$$

Step 3: Apply exponent rules to simplify the expression further

To simplify this expression further, we can now apply the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\): $$\frac{64 m^{-9} n^{18}}{8 m^{-15} n^3} = \frac{64}{8}\times m^{-9-(-15)}\times n^{18-3} = 8 m^{6} n^{15}$$ #Final_solution#Our simplified expression containing only positive exponents is: $$8 m^{6} n^{15}$$