How do you simplify (343 u^4 c^-5) /(7 u^6 c^-3)^-5?

Aug 8, 2018

$\frac{5764801 {u}^{34}}{{c}^{20}}$

Explanation:

There is a negative exponent rule, I'm not quite sure if it has a name, but it says that a negative exponent in the numerator can be moved to the denominator and become positive, and vice versa.
An example would be ${x}^{-} 2 = \frac{1}{x} ^ 2$

So using this

$\frac{{\left(7 {u}^{6} {c}^{-} 3\right)}^{5} \left(343 {u}^{4}\right)}{c} ^ 5$

Then we can distribute the exponent, $5$, in the numerator
$\frac{16807 {u}^{30} {c}^{-} 15 \left(343 {u}^{4}\right)}{c} ^ 5$

Now we can move the ${c}^{-} 15$ to the denominator using the negative exponent rule
$\frac{16807 {u}^{30} \left(343 {u}^{4}\right)}{{c}^{5} \cdot {c}^{15}}$

We can now combine like bases

$\frac{5764801 {u}^{34}}{{c}^{20}}$

Aug 9, 2018

${\left({x}^{a}\right)}^{b} = {x}^{a \times b}$

$\frac{343 {u}^{4} {c}^{-} 5}{7 {u}^{6} {c}^{-} 3} ^ - 5 = \frac{{7}^{3} {u}^{4} {c}^{-} 5}{{7}^{-} 5 {u}^{-} 30 {c}^{15}}$

${x}^{a} / {x}^{b} = {x}^{a - b}$

${7}^{8} {u}^{34} {c}^{-} 20$

or $\frac{{7}^{8} {u}^{34}}{c} ^ 20$

Aug 9, 2018

$\frac{{7}^{8} {u}^{34}}{{c}^{20}}$

Explanation:

$\frac{343 {u}^{4} {c}^{-} 5}{7 {u}^{6} {c}^{-} 3} ^ - 5$

Use the law of indices for negative indices:

${x}^{-} m = \frac{1}{x} ^ m$

$= \frac{343 {u}^{4} \times {\left(7 {u}^{6} {c}^{-} 3\right)}^{5}}{c} ^ 5$

Note that $343 = {7}^{3}$

It is better to keep the numbers in index form.

$= \frac{{7}^{3} {u}^{4} \times {7}^{5} {u}^{30} {c}^{-} 15}{c} ^ 5$

$= \frac{{7}^{3} {u}^{4} \times {7}^{5} {u}^{30}}{{c}^{5} \times {c}^{15}}$

Add the indices of like bases:

$= \frac{{7}^{8} {u}^{34}}{{c}^{20}}$