How do you simplify [a^(1/5) b^½ c^¼]÷[a^-½ b^2.5 c^2] ?

Apr 1, 2017

${a}^{\frac{7}{10}} / \left({b}^{2} {c}^{\frac{7}{4}}\right)$

Explanation:

$\left[{a}^{\frac{1}{5}} {b}^{\frac{1}{2}} {c}^{\frac{1}{4}}\right] \div \left[{a}^{- \frac{1}{2}} {b}^{2.5} {c}^{2}\right]$

Which is equal to:

$= \frac{{a}^{\frac{1}{5}} {b}^{\frac{1}{2}} {c}^{\frac{1}{4}}}{{a}^{- \frac{1}{2}} {b}^{2.5} {c}^{2}} = \frac{{a}^{\frac{1}{5}} {b}^{\frac{1}{2}} {c}^{\frac{1}{4}}}{{a}^{- \frac{1}{2}} {b}^{\frac{5}{2}} {c}^{2}}$

Which we can sort individually by base:

$= {a}^{\frac{1}{5}} / {a}^{- \frac{1}{2}} \left({b}^{\frac{1}{2}} / {b}^{\frac{5}{2}}\right) {c}^{\frac{1}{4}} / {c}^{2}$

We can simplify each of these using the rule ${x}^{m} / {x}^{n} = {x}^{m - n}$:

$= {a}^{\frac{1}{5} - \left(- \frac{1}{2}\right)} \left({b}^{\frac{1}{2} - \frac{5}{2}}\right) {c}^{\frac{1}{4} - 2}$

Finding common denominators:

$= \left({a}^{\frac{2}{10} + \frac{5}{10}}\right) \left({b}^{\frac{1}{2} - \frac{5}{2}}\right) \left({c}^{\frac{1}{4} - \frac{8}{4}}\right)$

$= {a}^{\frac{7}{10}} {b}^{- 2} {c}^{- \frac{7}{4}}$

Simplify this using the rule ${x}^{-} m = \frac{1}{x} ^ m$:

$= {a}^{\frac{7}{10}} \left(\frac{1}{b} ^ 2\right) \frac{1}{c} ^ \left(\frac{7}{4}\right)$

Which can be combined:

$= {a}^{\frac{7}{10}} / \left({b}^{2} {c}^{\frac{7}{4}}\right)$