# How do you simplify (13u²v(u^4v²w^7)³uw^6)/(-26(uv^4)³v³(uw²)²)?

Jun 3, 2015

Starting with $\frac{13 {u}^{2} v {\left({u}^{4} {v}^{2} {w}^{7}\right)}^{3} u {w}^{6}}{- 26 {\left(u {v}^{4}\right)}^{3} {v}^{3} {\left(u {w}^{2}\right)}^{2}}$

it looks like it's going to be easiest to total up the powers of $u$, $v$ and $w$ separately in the numerator and denominator, using the properties of exponents like:

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

Numerator:
$u : 2 + \left(4 \times 3\right) + 1 = 15$
$v : 1 + \left(2 \times 3\right) = 7$
$w : \left(7 \times 3\right) + 6 = 27$

Denominator:
$u : \left(1 \times 3\right) + \left(1 \times 2\right) = 5$
$v : \left(4 \times 3\right) + 3 = 15$
$w : \left(2 \times 2\right) = 4$

So the result of dividing numerator by denominator is:
$u : 15 - 5 = 10$
$v : 7 - 15 = - 8$
$w : 27 - 4 = 23$

The scalar term is just $\frac{13}{- 26} = - \frac{1}{2}$

Putting this all together we get:

$\frac{13 {u}^{2} v {\left({u}^{4} {v}^{2} {w}^{7}\right)}^{3} u {w}^{6}}{- 26 {\left(u {v}^{4}\right)}^{3} {v}^{3} {\left(u {w}^{2}\right)}^{2}}$

$= - \frac{{u}^{10} {w}^{23}}{2 {v}^{8}}$

with restrictions $u \ne 0$, $v \ne 0$, $w \ne 0$.